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, no matter how many times they come.
You must always see yourself on the other side of it. What you see is what you will get!
A wise man once asked his children: Can you say a professor is a failure? They replied, "NO! The father responded: Wole Soyinka failed the West African Senior Secondary Certification Examination nine (9) times⦠The children screamed! But can you still call him a failure today? No, the children replied, The father then asked why? Then, the children responded, "because he confronted his fears and was determined enough to overcome them by hard work and with the influence of divine grace⦠and eventually succeeded."
The moral of this illustration is that you are not a failure until you give up!
Today, you call him Prof. Wole Soyinka (a hero), but as at the time he was failing WAEC, most people called him a dullard (a zero).
Adversity is a ladder to greater levels of glory!
You will not get your reward until you finish your course!
You cannot get your crown until you are done with the cross!
No guts, no glory! You will not be able to overcome the wilderness until you become wild!
You cannot get through to the promised land until you are completely processed (refined)!
No gold ever shines without having passed through the furnace!
You cannot become more until you've been mocked!
You will not be able to receive your prizes until you have fully paid your price! What we regard as adversity is actually a setup for us to overcome and step up! It is a junction, or curve, on our journey to fulfill our destiny! Don't interrupt the processes of growth and sustainable development.
Adversity is an examination in disguise to build our muscles so as to be fit enough to carry or bear the weight of glory!
Therefore, adversity is:
YOU! You determine what must come out of it! Adversities are surmountable! You are the actor in the picture. I charge you to BE THE HERO!
You are not a failure until you give up! Adversity unveils us to realms of glory! It compels us to aim higher, while it charges us to take the stairs!
Dr. Martin Luther King Jnr. reveals that: 'The ultimate measure of a man is not where he stands in moments of comfort and convenience, but where he stands at times of challenge and controversy!'
It is not easy to be EASY. We must be determined, we must work and do our best to escape genuinely, but only God can deliver us from all uneasiness. Only God gives us the grace to overcome all odds!
Stop being frustrated when you are confronted with challenges. Rather, engage in audacious faith to confront and conquer your fears! It is a revival, and you are in for a revolution that will propel the required evolution. Adversity is an advanced citadel of learning in disguise! I call it the advanced 'universe's-city', where reality poses a great threat, just like examinations within the four walls of a university.
Thank you all for reading this #EpistleForChampions
Is Britain Still Qualified to Call Nigeria "Fantastically Corrupt"?
By Joel Popoola
With a Prime Minister under police investigation and Β£4.3bn of COVID-19 fraud, can Britain continue to call Nigeria "fantastically corrupt"?
The next time a member of the British establishment describes Nigeria as "a failed state" perhaps we should remind them of the time a government minister resigned after admitting to overseeing a Β£4 billion fraud β and it wasn't even the biggest political scandal that week!
In London, Prime Minister Boris Johnson β recently censured for accepting in the region of 62m Naira to refurbish his flat from a wealthy backer β stands accused of hosting boozy parties during COVID-19 lockdown.
One such incident β for which his secretary invited attendees to bring alcohol to a gathering in the Prime Minister's house β saw Johnson insist that he believed himself to have been at "a work event".
Johnson now stands accused of enjoying a birthday party during a time when up and down the UK β and around the world β parents were telling their children they were not allowed one. This was even a time when the Queen sat alone at her husband's funeral to obey social distancing requirements.
Had no-one
|
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| 0.881081 | -0.550915 | -0.669651 | 0.156592 |
told Johnson it was his birthday? Is him being presented with a cake and serenaded a regular workday occurrence?
The British Prime Minister is now subject to a police investigation.
I'll repeat that.
While all of this was going on, Finance Minister Lord Theodore Agnew stood up in Parliament to announce that no less that Β£4.3bn of public money set aside for COVID-19 assistance for businesses had been lost to fraud.
The admission came after the New York Times described the UK's pandemic spending as being characterised by "waste, negligence and cronyism", leading to a situation where "politically connected businesses reaped billions".
The American newspaper has estimated that "about half" of the UK's pandemic spending went to "companies with political connections, no prior experience or histories of controversy."
These included contracts for one company currently on the receiving end of two global corruption probes, and a $470m protective equipment contract to a pest control firm who supplied 600,000 unusable face masks.
At least Boris Johnson has not publically accused Nigeria of being "fantastically corrupt" as one of his predecessors did!
Lord Agnew β the government minister with the responsibility for fighting financial fraud! β admitted that "schoolboy errors" saw 1000 "ghost" businesses given public money.
UK government figures suggest that overall Β£5.8 billion was stolen from pandemic relief schemes by people claiming cash they weren't entitled to.
Again, imagine the response in Britain if a Nigerian politician admitted this!
But here perhaps Nigerian leaders could learn something from their British counterparts.
Lord Agnew was not personally to blame for this fraud β presumably he did not order his officials to allow it β but he accepted responsibility for it and immediately tendered his resignation.
This prompted praise from opposition parliamentarians, with one calling him "a minister who felt his integrity could no longer ensure he remained a member of the Government.
"Can I just take this opportunity to say on behalf of these benches how much we appreciate the honour and integrity that has just been displayed by the minister", said another.
This shouldn't need saying, but people appreciate openness and transparency from their leaders.
At the digital democracy campaign I lead, we aim to give them the platforms they need to deliver it.
We have created a free mobile app called Rate Your Leader, which was designed to reconnect electors and the elected, opening direct channels of communication between people and their elected officials β giving local people the kind of access previously only enjoyed by funders.
Rate Your Leader encourages politicians to speak directly to the people they serve and explain the decisions they have made and the reasons for them. If the voters don't like the answer they get, they can rate their politicians badly.
This leads to greater levels of trust in a political class that the voters can see are working for them, and accountable to them.
Digital technologies like Rate Your Leader put transparency and accountability and your fingertips. Direct communication from politician to person, peer to peer.
We'd be happy to offer Boris Johnson a free account. But we must warn him, if he continues to break the rules he made β in the rooms he made them β he is unlikely to be rated very highly.
Joel Popoola is a Nigerian tech entrepreneur, digital democracy campaigner and creator of the Rate Your Leader app. He can be reached via @JOPopoola
Our Helicopter Did Not Crash-land, No One Sustained Injuries β Police
By Eric Elezuo
The Nigerian Police have refuted reports of a crash involving one of its helicopters, saying that what was experienced was a controlled safe landing where all the six passengers and crew on board came out unhurt.
A statement from the Force Headquarters, Abuja, and signed by the Force Public Relations Officer, CP Frank Mba, noted that the aircraft, "which was flown by one of the best Police Pilots, was professionally safe-landed with minor damage on the rear rotor as a result of obstacle at the landing spot."
Below is the detailed statement:
The Nigeria Police Helicopter, Bell 429 5NMDA, flying from Abuja to Bauchi on Wednesday, January 26, 2022, was involved in a controlled safe landing at the Sir Abubakar Tafawa
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CommonCrawl
| 0.58628 | -0.193213 | 0.773894 | 0.199598 |
4/11/19 Nashville, TN Renaissance Nashville Scott MacIntyre performs for AMSURG's Business Office Conference.
3/29/19 Toledo, OH Christ The King School FREE Scott MacIntyre performs for Christ The King School.
3/26/19 Sevierville, TN Sevierville Convention Center Scott MacIntyre performs for Sevierville Chamber of Commerce Annual Membership Breakfast.
3/19/19 Nashville, TN Tennessee State Capitol Scott MacIntyre performs for TN Home Educators "Rally Day".
3/9/19 Laguna Beach, CA Ritz Carlton Laguna Niguel Scott MacIntyre performs for ACU President's Firestorm Summit.
1/24/19 Smyrna, TN Lancaster Christian Academy Scott MacIntyre performs for "Spiritual Emphasis Week."
9/7/13 Smyrna, TN Lancaster Christian Academy Scott MacIntyre performs for "Middle School Retreat Day."
6/10/18 Reedley, CA Reedley Mennonite Brethren Church FREE Scott MacIntyre performs for morning services.
6/9/18 Reedley, CA Sierra View Homes Scott MacIntyre in concert at Sierra View Homes.
4/23/18 Oakville, ON Kings Christian Collegiate Scott MacIntyre performs for morning chapel.
4/21/18 Toronto, ON Scott MacIntyre performs for Peel Catholic District School Board.
1/19/18 El Cajon, CA Holy Trinity School Scott MacIntyre performs for morning chapel.
12/15/17 Scottsdale, AZ Phoenix Seminary Scott MacIntyre performs for the 2017 Phoenix Seminary Christmas Party.
11/30/17 Old Bridge, NJ Grand Marquee Scott MacIntyre key-note for Hackensack Meridian Health's Mitchell Vasser Vision Awareness Day.
11/28/17 Phoenix, AZ Scottsdale Christian Academy Scott MacIntyre performs for morning chapels.
11/17/17 Nashville, TN Sheraton Music City Scott MacIntyre key-note for Tennessee Valley Public Power Association.
5/17/17 Phoenix, AZ Phoenix, AZ Scott MacIntyre performs for the Arizona Diamondbacks versus New York Mets game.
12/30/16 Los Angeles, CA Scott MacIntyre performs for the Donate Life "Rose Parade Floragraph Brunch."
12/29/16 Los Angeles, CA Scott MacIntyre performs for the Donate Life "Rose Parade Rider/Walker Dinner."
11/9/16 Franklin, TN Franklin Christian Academy Scott MacIntyre performs for morning chapel.
10/18/16 Cincinnati, OH Crossroads Mason Scott MacIntyre keynote speech for the Luxottica Partner Innovation Summit.
10/1/16 Lexington, KY Embassy Suites Hotel Scott MacIntyre keynote speech for Kentucky Organ Donor Affiliates.
4/26/16 Nashville, TN Opryland Resort Scott MacIntyre in concert for Raymond James Financial Group.
4/16/16 Pleasanton, CA Alameda County Fairgrounds Scott MacIntyre performs for Donor Network West.
4/15/16 Los Angeles, CA Scott MacIntyre keynote speech for Donate Life Blue & Green Day.
4/14/16 Los Angeles, CA Scott MacIntyre keynote speech for the 2016 Donate Life Bowling Bonanza.
4/10/16 Phoenix, AZ Chase Field Scott MacIntyre performs the National Anthem for the Arizona Diamondbacks.
2/19/16 Murfreesboro, TN Middle Tennessee State University Scott MacIntyre keynote speech for university students.
1/3/16 Scottsdale, AZ Scottsdale Bible Church FREE Scott MacIntyre performs for morning services.
1/2/16 Scottsdale, AZ Scottsdale Bible Church FREE Scott MacIntyre performs for evening service.
12/6/15 Saskatoon,
|
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] | 1,024 |
C4
| -1.259963 | -1.434707 | 0.106815 | -1.315725 |
SK The Neighborhood Church $20 Scott MacIntyre performs for The Neighborhood Church "Christmas Banquet." Contact the church for MORE INFO.
12/6/15 Prince Albert, SK Crossroad Church FREE Scott MacIntyre performs for morning services.
11/14/14 San Diego, CA Scott MacIntyre in concert at the 13th Annual Pro Athletes for Life Gala.
9/27/14 Anaheim, CA Disneyland Resort Scott MacIntyre in concert for the Spirit of Hope Gala with Hope International University.
8/16/14 Hanoi, Vietnam Press Club Hanoi Scott MacIntyre performs for Viet UC Family.
8/9/14 Ho Chi Man City, Vietnam Gala Royal Scott MacIntyre performs for Viet UC Family.
8/6/14 Phoenix, AZ Chase Field Scott MacIntyre sings the National Anthem at the Diamondbacks vsrnKansas City Royals game.
7/30/14 San Francisco, CA Met Life Stadium Scott MacIntyre performs the National Anthem at the Giants vs Pirates game.
7/28/14 San Francisco, CA Mortons Scott MacIntyre performs for the World Transplant Conference.
5/3/14 Antioch, TN Glencliff High School Scott MacIntyre at Little Kids Rock "Jam Summit."
4/26/14 Washington DC National Portrait Gallery Scott MacIntyre MC's and performs for Learning Ally's 54th annual National Achievement Awards gala.
3/30/14 Nashville TN Bridgestone Arena Scott MacIntyre performs the National Anthem for the Predators vs Capitols NHL game.
10/19/13 Las Vegas, NV Westin Las Vegas Scott MacIntyre performs in support of Nevada Donor Network and the Simon Keith Foundation.
2/1/13 Queen Creek, AZ Edu-prize School Scott MacIntyre performs for National School Choice Week closing ceremonies.
1/25/13 Phoenix, AZ Phoenix Convention Center Scott MacIntyre performs with the Jonas Brothers for National School Choice Week Rally.
1/24/13 Anaheim, CA Anaheim Convention Center Scott MacIntyre performs at Winter NAMM 2013 in support of Sauter Pianos.
12/2/12 Loma Linda, CA Loma Linda University Scott MacIntyre in concert for Loma Linda University Medical Center.
11/10/12 Phoenix, AZ Mayo Clinic Scott MacIntyre performs for Mayo Clinic Hospital annual transplant banquet.
10/27/12 Phoenix, AZ Mayo Clinic Scott MacIntyre performs for Mayo Clinic Hospital annual transplant banquets.
10/24/12 Scottsdale, AZ Scottsdale Bible Church Scott MacIntyre key-note speech for national Community Bible Study program.
10/15/12 Scottsdale, AZ Steinway Pianos $5 Scott MacIntyre key-note speech and book signing for Brandeis National Committee Meeting. Admission $5 at door.
10/8/12 Nashville, TN Nashville Performing Arts Center Scott MacIntyre presents the IEBA Career Achievement Award to Paula Abdul.
10/4/12 Dallas, TX Gaylord Hotel Scott MacIntyre keynote speech for National Learning Congress.
9/29/12 Anaheim, CA Disneyland Hotel Scott MacIntyre performs for Hope International University's "Spirit of Hope" gala.
9/28/12 Pittsburgh, PA PNC Park Scott MacIntyre performs the national anthem for Pirates vs Reds baseball game in support of Donate Life America.
9/28/12 Pittsburgh, PA CAPA Highschool Scott MacIntyre performs for CAPA Highschool assembly.
04/20/12 La Jolla, CA San Diego Marriott La Jolla Scott MacIntyre key-note speech for Sjogrens Syndrome Foundation national conference.
04/18/12 Scottsdale, AZ Barnes & Noble Scott MacIntyre signs his book "By Faith, Not By Sight" at the Barnes
|
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] | 1,024 |
C4
| -1.056183 | -1.58525 | 0.080813 | -1.310665 |
Q: Async Tcp Server cannot receive data I am currently tyring to create a Multi threading and async tcp server that implements the Tcpl listener
Current the server is working as intended as i am able to send data to the server an transmit data to the client with out any problems
However after i have sent data to the server and then sent data back to the client, when the client sends data back to the server again the server is unabe to pick up the data
I have tried for days to find an answer to this problem however with no luck
Here is the code that i am currently using in the server:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Net;
using System.Net.Sockets;
using System.Threading;
using System.Windows.Forms;
using System.IO;
namespace MyTcpAsyncClass
{
public class StateObject
{
public TcpClient MyTcpClient = null;
public NetworkStream MyNetworkStream = null;
public const int MyBufferSize = 1024;
public byte[] MyBuffer = new byte[MyBufferSize];
public string RequestString = "";
public StringBuilder MyStringBuilder = new StringBuilder();
char[] RequestChars; // Char array of Request
const char STX = (char)0x02; // Start Character
const char FTX = (char)0x03; // Finish Character
public void Dispose()
{
try
{
MyTcpClient.Close();
MyNetworkStream.Close();
MyNetworkStream.Dispose();
}
catch (Exception ex)
{
MessageBox.Show("Message:\n" + ex.Message + "\n\nStacktrace:\n" + ex.StackTrace);
}
}
}
public static class AsyncServerFunctions
{
private static int mPort = 0;
private static ManualResetEvent MyManualResetEvent = new ManualResetEvent(false);
public static void StartListening()
{
//Catch to Tcp Client Connection
try
{
//Get the database connection
//MyReaderWriterLockSlim.EnterReadLock();
LoadSettings();
//MyReaderWriterLockSlim.ExitReadLock();
TcpListener MyTcpListener = new TcpListener(IPAddress.Any, mPort);
MyTcpListener.Start();
while (true)
{
//Set the event to nonsignaled state
MyManualResetEvent.Reset();
//Start an asynchronous TcpListener to listen for a connection
MyTcpListener.BeginAcceptTcpClient(AcceptTcpClientCallback, MyTcpListener);
//Wait until a connection is made before continuing
MyManualResetEvent.WaitOne();
}
MyTcpListener.Stop();
}
catch (Exception ex)
{
AddErrorLog(ex.Message, ex.StackTrace);
}
}
private static void AcceptTcpClientCallback(IAsyncResult result)
{
try
{
//BeginAcceptTcpClientCallback
//Signal the main thread to continue
MyManualResetEvent.Set();
//Get the TcpClientNetworkStream:
TcpListener MyTcpListener = (TcpListener)result.AsyncState;
//Finish Async Get Client Process
TcpClient MyTcpClient = MyTcpListener.EndAcceptTcpClient(result);
StateObject MyStateObject = new StateObject();
MyStateObject.MyTcpClient = MyTcpClient;
MyStateObject.MyNetworkStream = MyTcpClient.GetStream();
//Begin Async read from the NetworkStream
MyStateObject.MyNetworkStream.BeginRead(MyStateObject.MyBuffer, 0, StateObject.MyBufferSize, new AsyncCallback(BeginReadCallback), MyStateObject);
}
catch (Exception ex)
{
AddErrorLog(ex.Message, ex.StackTrace);
}
}
private static void BeginReadCallback(IAsyncResult result)
{
StateObject MyStateObject = (StateObject)result.AsyncState;
NetworkStream MyNetworkStream = MyStateObject
|
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] | 1,024 |
StackExchange
| -1.800658 | -0.087606 | -1.612887 | -0.87618 |
.MyNetworkStream;
string MyRequestString = "";
try
{
//Get Request Data here
if (MyStateObject.MyBuffer.Length > 0)
{
//Store the data recived
MyStateObject.MyStringBuilder.Clear();
MyStateObject.MyStringBuilder.Append(Encoding.ASCII.GetString(MyStateObject.MyBuffer));
//Get the stored Request string
MyRequestString = MyStateObject.MyStringBuilder.ToString();
//Record the string recived
DatabaseFunctions.AddMessageLog("String Recived (BeginReadCallback): " + MyRequestString);
//Remove the first and last character
MyRequestString = CleanString(MyRequestString);
//Record the Request String
DatabaseFunctions.AddMessageLog("Request String Recived:" + MyRequestString);
//Get the Message Identifier
string MessageIdentifier = "";
MessageIdentifier = MyRequestString.Substring(0, 2);
switch (MessageIdentifier)
{
case "value":
SendResponse(MyStateObject, StartUp(MessageIdentifier, MyRequestString));
SendResponse(MyStateObject, SendTransactionStart(MessageIdentifier, MyAmount));
GetResponse(MyStateObject);
break;
default:
//***Default Case***
SendResponse(MyStateObject, DefaultCase(MyRequestString));
break;
}
//Dispose of the connection
MyStateObject.Dispose();
}
}
catch (Exception ex)
{
AddErrorLog(ex.Message, ex.StackTrace);
try
{
MyStateObject.Dispose();
}
catch
{
AddErrorLog(ex.Message, ex.StackTrace);
}
}
}
private static void SendResponse(StateObject pMyStateObject, string pResponseString)
{
try
{
//Send a response to the client
//Get bytes from string sent
byte[] MyResponseBytes = Encoding.ASCII.GetBytes(pResponseString);
//Get the network stream
NetworkStream MyNetworkStream = pMyStateObject.MyNetworkStream;
//Call SendResponseCallback
MyNetworkStream.BeginWrite(MyResponseBytes, 0, MyResponseBytes.Length, new AsyncCallback(SendResponseCallback), pMyStateObject);
}
catch (Exception ex)
{
AddErrorLog(ex.Message, ex.StackTrace);
}
}
private static void GetResponse(StateObject pStateObject)
{
//This will run a new AsyncCallback To get the response from the client
NetworkStream MyNetworkStream = pStateObject.MyNetworkStream;
pStateObject.MyBuffer = new byte[1024];
MyNetworkStream.BeginRead(pStateObject.MyBuffer, 0, pStateObject.MyBuffer.Length, new AsyncCallback(BeginReadCallback), pStateObject);
}
private static void SendResponseCallback(IAsyncResult result)
{
try
{
//End the send procedure
StateObject MyStateObject = (StateObject)result.AsyncState;
NetworkStream MyNetworkStream = MyStateObject.MyNetworkStream;
MyNetworkStream.Flush();
}
catch (Exception ex)
{
AddErrorLog(ex.Message, ex.StackTrace)
}
}
private static void ShowExceptionMessage(string pMessage, string pStacktrace)
{
MessageBox.Show("Message:\n" + pMessage + "\n\nStacktrace:\n" + pStacktrace);
}
private static void AddErrorLog(string pMessage, string pStackTrace)
{
DatabaseFunctions.AddMessageLog("Message:" + pMessage + "; Stacktrace:" + pStackTrace);
}
}
}
Thanks All
A: You should call BeginAcceptTcpClient in AcceptTcpClientCallback also. You don't accept any new connection after the first one.
A: In your BeginReadCallback function you dispose off the object you have used to invoke BeginRead try running the code without dispose function and
|
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] | 1,024 |
StackExchange
| -1.691976 | 0.061639 | -1.519639 | -0.978002 |
Back-It (1986) 0 Utility DOS
Back-It, from Gazelle Systems, is a fast and compact hard disk backup utility. It was sometimes part of OEM system bundles. It was kind of low-end, and changed backup formats between releases but otherwise did what it was supposed to do.
Backup Exec (1993) 2 Utility Windows DOS
Backup Exec is an easy to use backup program. It can back up files to floppy disks or certain tape drives. companies including Maynard Electronics, Archive Corp, Quest Development Corporation, Conner Peripherals, Arcada Software, Seagate Technology, Veritas Software, and Symantec. On top of that, Microsoft bundled a version with MS-DOS 6.
1.0 (Win)
BeckerTools (1991) 0 Utility Windows
BeckerTools is a set of disk utilities that competed with Norton Utilities and PC-Tools. Version 2.0 Plus for Windows includes a file manager shell, a disk editor, a disk checking tool, a backup utility, a disk defragmenter, and a screen saver.
Bitstream FaceLift (1990) 0 Utility Windows
Bitstream FaceLift works with Bitstream scalable typefaces to bring high quality type to Microsoft Windows applications instantly and easily for screen display, dot matrix printers, and laser printer output.
Bitstream TrueType Fontpack (1992) 0 Word Processor Publishing Windows
For every Windows 3.1 user! Forty versatile text and decorative fonts in TrueType format that you can scale to any size for your screen and printer - true WYSIWYG memos, and reports to newsletters, brochures, and invitations.
Borland Office (1994) 1 Word Processor Spreadsheet Presentations Database Windows
Borland Office is an office suite published by Borland built around WordPerfect, Paradox, and Quattro Pro. It competed unsuccessfully against Microsoft Office. It was later acquired by Novell and renamed "PerfectOffice", and then later became "Corel Office".
BULL Micral Prologue (1984) 0 DOS
BULL Micral Prologue system for the Olympia People Computer.
CA-Compete (1991) 0 Spreadsheet Financial Windows
CA-Compete! is a graphical, object-based, multi-dimensional modeling and data viewing tool that is as easy to use as a spreadsheet and as powerful as a decision support system (DSS) or executive information system (EIS). Compete! looks and works like a spreadsheet, but is object-based. In other words, you define the names that Compete! uses to identify and locate data in a model. A Compete! model can also contain many more dimensions than the spreadsheet norm of two or three.
Carbon Copy (1990) 0 Utility DOS Windows
Carbon Copy is a remote control desktop program for DOS and later Windows, similar to PCAnywhere. protection. You can not connect to other copies of Carbon Copy if the serial numbers are the same.
CD Creator (1996) 1 Utility Windows
Adaptec CD Creator, today sold by Roxio, was a popular a tool for mastering and burning CD images. It was targeted primarily at the home and office markets and often bundled with CD-Burner hardware. Some OEM versions were customized for specific drives. Adaptec also sold a similar, but different, program for Apple Macintosh computers under the name Toast.
1.x/2.x
Central Point Anti-Virus (1992) 0 Utility DOS
Central Point Anti-Virus was a DOS-based antivirus program developed originally by an Israel company, CARMEL Software Engineering Ltd. as "Turbo Anti-Virus", and licensed by Central Point Software Inc. It was the basis for Microsoft's Anti-Virus for DOS and Windows (MSAV and MWAV). In 1994 it was acquired by Symantec Corporation and merged into Norton Antivirus.
Central Point Backup (1992) 0 Utility Windows
A powerful and friendly backup utility from Central Point Software, that supports backing up to either floppy disks or tape drives. Central Point Backup features ease of use, speed, detailed backup history, and virus scanning. This product was more commonly found bundled as part of PC-Tools.
Change Directory
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] | 1,024 |
CommonCrawl
| -1.202905 | -0.561622 | 0.289723 | -0.34841 |
"I have been receiving John's work regularly for the last 6 months and have been astounded at how much better my body and mind function. His techniques unite and balance physical, emotional, mental and spiritual energies. I feel relaxed and energized simultaneously. John's love and enthusiasm abound and his work is a reflection of them both. I am extremely grateful to receive this work. Thank you brother John."
"John's clarity and integrity as a health facilitator are profound. I whole-heartedly endorse the work this fine man brings to this blessed community. John's working knowledge of the physical and etheric aspects of the human experience render him an asset to any individual seeking wellness!"
"When John asked if I wanted an Auric Polarization, I wasn't quite sure what it was, but I knew it had something to do with our Aura and Life Force, so I agreed to have one!!! Low and behold it has changed my whole life. Not only has my stress level been brought down, it brought back the feeling for a purpose in my life. After we were through I asked John to teach me this great art. John has become a great teacher to me, I feel like I walked away from classes with a great deal more than learning four disciplines. I found a career in which I can work, teach and write. I will be ever grateful, "
As we grow and change on our Spiritual Journeys, we realize that it is necessary to release physical, mental, emotional and spiritual blockages in order to progress to the next level of Spiritual Consciousness. There are many methods available to us that can aid us in these releases. It is not necessary to go through torture in order to awaken blocked responses so they can be dealt with, forgiven, and released through the heart. Once released from our self-imposed prisons of the past, we are filled with more Light, and we are able to create, with our minds and hearts united, new paradigms of thought. It is thinking that is not based upon our past and the blockages that kept us there believing in dis-ease, death, lack of abundance and pain; but rather, bodies filled with Love, Life, Light, Joy Health, Abundance and Peace. Auric polarization is a way in which one can release blockages within one's self.
These methods are simple ancient techniques. They relieve stress and tension, and are extremely relaxing and rejuvenating.
These techniques are non-invasive and do not require the removal of any clothing other than your shoes. I do not do massage.
After a session we recommend a ten or fifteen minute mediation each day to help in the releasing process. Close your eyes and go into your heart. Listen to yourself breathe. Breathe deep from your diaphragm, your stomach expanding and contracting as you breathe. Now count to seven as you breathe in, hold the breath for the count of seven and breathe out counting to seven. Once you exhale, hold the breathe in, repeating the process. This is called a complete circular breath.
Think only of this breath as you meditate. Each breath is filling you with Light, Love, Peace, Joy, Health and Abundance. Let your thoughts float by, giving no particular attention to any one thought. Try this, it will bring results.
Auric Polarization, developed by Reiki Master Teacher John Van Crump, employs the currents that naturally flow through our hands. By connecting with another's current of Life-Force, blockages can be released.
When the energy of Life-Force is flowing freely, we experience Joy, Love, Peace, Health and Abundance.
Alignment of the Electro-magnetic Field or Aura, takes place through physical and non-physical techniques, re-establishing the proper flow of Life-Force throughout the entire body, and balancing the right and left hemispheres of the brain.
These techniques relieve stress and bring deep relaxation to the body, while at the same time the cells in the body are charged with Life-Force, and the body is strengthened.
The four disciplines or techniques used in an Auric Polarization are Shiatsu (acupressure), Reiki (energy work), Reflexology (foot massage), and Polarization therapy.
These four disciplines combine to release stress and trapped energy on all four levels (physical, mental, emotional, and spiritual).
Bringing stress levels down frees us form potential dis-
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] | 1,024 |
C4
| 1.182675 | -0.143411 | -0.80235 | 0.53985 |
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|
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] | 1,024 |
C4
| -1.751751 | -2.184826 | -1.713307 | -2.564158 |
Only Believe by Gordon Robertson
A message from Gordon Robertson, speaking at CBN.
August 13, 2018 @ 7:16 pm
Releasing the Power of God by TB Joshua
Amalgamation of 3 sermons by TB Joshua (SCOAN), on Belief in the Heart, Hindrance to Prayer and the Power of God.
Generational Curses through lineage - Dr Francis Myles
Interview by Sid Roth with Francis Myles on generational curse through lineage.
Some examples: Certain sin impacts 10 generations (Deut. 23:2). Those who rebel against God because of unbelief, worship idols β this impacts 4 generations (Ex. 20:5; Num. 14:12). Sin in general impacts other generations (Ezek. 18:1-3) but the punishment of sin is only towards the individual (Deut. 24:16, Ezek. 18:20). A father who is put into jail for his sin impacts the family left behind. "Our fathers sinned and are no more, but we bear their iniquities" (Lam. 5:7).
Man was made in God's image (appearance), according to His likeness (thinking β Good - desires and wants) (Gen. 1:26). But man changed it (corrupted it), no longer according to God's image & likeness but after Adam's β to rebel against God (Gen. 5:3). It's in the genes. We need to forsake our spiritual lineage and adopt Christ's lineage. "He was taken from prison and from judgment, and who will declare His generation (Isa 53:8)" His Lineage? Yes we must declare His Lineage, not our own β denounce our allegiance to our ancestry like Paul did (Phil. 3:4-7) like Zacharias did for his son John the Baptist (Luke 1:59-63).
Yeshua broke the curse by being cursed for us (Gal. 3:13). It is a covenant (legal contract) of exchange, we are made into the image of Christ (Rom. 8:29). The blessings of God are on 1000 generations of those who love God and keep His commandments (Deut. 7:9). Yeshua enables us to keep His commandments for He is with us (Mat. 1:23) and Holy Spirit is in us (Eph. 1:13).
Body Parts in Heaven - Gary Wood
Gary Wood dies in a car accident, goes to heaven and tells of the things God has instore for those who believe in Yeshua (Jesus). This is part of the 10 top interviews that moved me by Sid Roth.
The Supernatural by Ryan Wyatt
Ryan Wyatt a guest on Sid Roth shares his life of the supernatural because of Jesus. This is part of the 10 top interviews that moved me by Sid Roth.
Money the most important thing in Life? Story of a man Called Steve
Short Story of a man called Steve.
This is a story about a boy called Steve. His parents died when he was young and at the age of 9 years old he moved from one foster parent to the next. He was a tough kid, never shed a tear and refused to. It was for the weak. He looked for peace, true peace and love but couldn't find it. At the age of 12 he read a book by called Money, and said, "This is the most important thing in this life", he dedicated the next 10 years of his life pursuing more money through education. At the age of 29 he invented a highly resistant to weather car paint. This was revolutionary and had many car buyers competing for his patent. At the age of 31 he was a Billionaire β He bought anything his heart desired - houses, cars and had big parties. One day after his party, sitting on his porch overlooking his fields his heart was still heavy. He thought that money was the key to life but yet he felt empty. So, he decided to visit the neighborhood where he grew up as an observer. He down dressed and bought an old broken car. He already received many letters from people who he grew up with begging for money and didn't want them to recognize him as he
|
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CommonCrawl
| -0.362653 | -0.236851 | -0.563851 | -0.306669 |
drove through the streets. One evening just before sunset as he took a drive through the streets, he stopped outside one of the houses. He noticed a family sitting around a table to eat. They had nothing really on the table yet they bowed their head to pray, when done they started eating. They seemed happy. Was money really the key to peace? He had lots of it and everything seemed so fake. His friends fake, the women he knew were fake, even the stuff he had was β¦. Bla - fake, just stuff in a toy chest.
Getting ready for a shower, he noticed a red swollen mark under his armpit. He never noticed this before. He called the doctor who quickly had him go for the best doctors and examinations that money could buy. The doctors confirmed his fear β Lymphoma Cancer. Fear set in and Steve didn't know who to turn to, his money couldn't help him. As he started his 2nd round of Chemo β he thought, if he could he would give all my money to be healthy again. So, health was more important than money. But why wasn't he satisfied when he had the health. In one of the Chemo sessions as he sat in the waiting area, a Christian came up to him gave him a New Testament which Steve put into his pocket. The man proceeded to tell Steve about Yeshua, God's love for him and that surrendering to Yeshua is the answer to true peace. Just then Steve snapped and swore at the Christian, telling him to bug someone else and this was a fairy tale. After Steve had completed the 2nd round of Chemo and further testing, the doctor called Steve into his office and told him that he had less than 1 year to live. Steve went home, switched off the phone, moped around and isolated himself from friends and family. One morning, after 10months, he decided to leave the house and walk to the nearby park. On the way, a deep red rose caught his eye. He studied this rose, he couldn't make this rich textural colour in all of his paint shops. It was beautiful, just then a bee flew out, his wings glistening from the sun. He looked at all the flowers and trees on the way to the park. He forgot that he was sick because he enjoyed the very moment. As he stood at the park pondering on why he felt good, he realised that enjoying the moment, the time was precious. And if he savoured all the time in moments, he would have been more fulfilled. So enjoying the moment was more important than Health, was more important than Money. He sat down on a nearby park bench and couldn't help thinking, if he lived every day and moment to the fullest, he would still die. The future of the unknown troubled his heart. He picked a flower and put it into his jacket pocket. It brushed against a book. He pulled the book, it was the Bible the young man gave him. He flipped open and started reading. The title read "Yeshua in Heaven waiting for you John 14" and proceeded.
Was this a coincidence that he arrived on the very same passage that called out "Let not your heart be troubled" He read further "Ye believe in God, believe also in me. In my Father's house are many mansions: if it were not so, I would have told you. I go to prepare a place for you. And if I go and prepare a place for you, I will come again, and receive you unto myself; that where I am, there ye may be also." Was this true, was there a place beyond death. He turned to a bookmarked page the young man had obviously placed β John 3. The scripture was highlighted "For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life." And continued to read until evening. He went down on his knees and repented for rejecting God and His Son. He confessed that He surrendered his life to Yeshua and needed his help. At that moment peace and joy filled his heart that he couldn't explain. He got up saying Hope in the Eternal, is greater than the moment, is greater than health and greater than money. Walking home thinking on the events of the day, he looked up and spoke to God as his Heavenly father. Tears rolled down his cheeks as he realised there was some greater than hope β it was knowing and receiving the love of God. The final 2 months Steve was full of life and his joy infected the people
|
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CommonCrawl
| 0.959876 | -0.459746 | -0.646339 | 0.112321 |
Em ciΓͺncia, energia (do grego ΞΞ½, "dentro", e Ξ΅ΟΞ³ΞΏΞ½, "trabalho, obra"Λ ou seja, "dentro do trabalho") refere-se a uma das duas grandezas fΓsicas necessΓ‘rias Γ correta descrição do inter-relacionamento - sempre mΓΊtuo - entre dois entes ou sistemas fΓsicos. A segunda grandeza Γ© o momento. Os entes ou sistemas em interação trocam energia e momento, mas o fazem de forma que ambas as grandezas sempre obedeΓ§am Γ respectiva lei de conservação. A energia Γ© uma grandeza escalar que tem por grandeza conjugada o tempo; ao passo que o momento Γ© uma grandeza vetorial que tem por grandeza conjugada o vetor posição. Um ente fΓsico Γ© essencialmente caracterizado pela sua relação de dispersΓ£o, a relação entre energia e momento do ente.
Introdução
Γ bem difundido - nΓ£o sΓ³ em senso comum - que energia associa-se geralmente Γ capacidade de produzir um trabalho ou realizar uma ação. Em verdade, a etimologia da palavra tem origem no idioma grego, onde Ξ΅ΟΞ³ΞΏΟ (ergos) significa "trabalho". Embora nΓ£o completamente abrangente no que tange Γ definição de energia, esta associação nΓ£o se mostra por completo fora do domΓnio cientΓfico, e, em princΓpio, qualquer ente que esteja a trabalhar - por exemplo, a mover outro objeto, a deformΓ‘-lo ou a fazΓͺ-lo ser percorrido por uma corrente elΓ©trica - estΓ‘ a "transformar" parte de sua energia, transferindo-a ao sistema sobre o qual realiza o trabalho.
A histΓ³ria da energia comeΓ§a com o termo grego αΌΞ½ΞΟγΡια (energeia), possivelmente cunhado no ambiente da Academia por seus membros, ou no contexto da teoria mΓ©dica hipocrΓ‘tica ou por AristΓ³teles, com a ocorrΓͺncia mais antiga preservada hoje aparecendo nas obras deste em sua teoria de potencialidade e atualidade. A palavra foi utilizada por Kepler em 1619, mas aparece em sentido moderno provavelmente pela primeira vez no uso por Johann Bernoulli, que a emprega para indicar equilΓbrio de forΓ§as virtuais. Leibniz empregaria a definição de energia cinΓ©tica sob a denominação "vis viva", em controvΓ©rsia contra os cartesianos. Γ com o desenvolvimento da termodinΓ’mica, em meados do sΓ©culo XIX, que ela adquire sentido fundamental na ciΓͺncia, naquele contexto vinculado Γ quantidade de trabalho economizado e forΓ§a. Na dΓ©cada de 1850, William Thomson e William Rankine substituirΓ£o a antiga linguagem da mecΓ’nica com novos termos como "energia atual", "energia cinΓ©tica" e "energia potencial".
O conceito de energia Γ©, como tal, um dos conceitos essenciais da fΓsica. Em sua definição atual, nascida no sΓ©culo XIX, desempenha papel crucial nΓ£o sΓ³ nesta Γ‘rea do conhecimento, mas tambΓ©m em outras Γ‘reas da ciΓͺncia que, todas juntas, integram a ciΓͺncia moderna. Notoriamente relevante tanto na quΓmica quanto na biologia, e mesmo em economia, e outras Γ‘reas de cunho social, a energia se destaca como um ponto fundamental, uma vez que o comΓ©rcio de energia move anualmente quantidades enormes de dinheiro.
Pela sua importΓ’ncia, hΓ‘, na fΓsica, uma subΓ‘rea dedicada quase que exclusivamente ao estudo da energia: a termodinΓ’mica. Em termodinΓ’mica, o trabalho Γ© uma entre as duas possΓveis formas de transferΓͺncia de energia entre sistemas fΓsicos; a outra forma Γ© o calor.
Definição cientΓfica
O conceito cientΓfico de energia sΓ³ pode ser entendido mediante a anΓ‘lise de dois entes ou sistemas fΓsicos em interação. Quando dois sistemas fΓsicos interagem entre si, mudanΓ§as nos dois sistemas ocorrem. A interação
|
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Wikipedia
| 0.374349 | 1.023298 | -0.649253 | 0.74223 |
entre sistemas fΓsicos naturais dΓ‘-se, em acordo com os resultados empΓricos, sempre de forma muito regular, sendo uma mudanΓ§a especΓfica em um deles sempre acompanhada de uma mudanΓ§a muito especΓfica no outro, embora estas mudanΓ§as possam certamente ser de naturezas muito ou mesmo completamente distintas.
Regularidades observadas na natureza expressam-se dentro da ciΓͺncia mediante o estabelecimento das denominadas leis cientΓficas. No que se refere Γ forma com que dois entes fΓsicos interagem entre si, na busca da correta correlação entre as mudanΓ§as observadas nos sistemas viu-se a necessidade de se estabelecerem, para o correto cumprimento da tarefa, nΓ£o apenas uma mas duas grandezas fΓsicas primΓ‘rias independentes, cada qual associada a uma lei de conservação prΓ³pria, leis estas inerentes a todos os sistemas fΓsicos e que combinadas, permitem a correta descrição dos mesmos. Tais grandezas fΓsicas sΓ£o denominadas energia e momento linear, e as leis cientΓficas que as governam denominam-se respetivamente lei da conservação da energia e a lei da conservação do momento linear. Ao passo que o momento Γ© uma grandeza vetorial, a sua contraparte aqui descrita Γ© uma grandeza escalar.
Γ relação existente entre a energia e o momento de um dado ente fΓsico, dΓ‘-se o nome de relação de dispersΓ£o, sendo, esta, vital no contexto de qualquer teoria para a dinΓ’mica da matΓ©ria e energia (mecΓ’nica clΓ‘ssica, relatividade, mecΓ’nica quΓ’ntica etc.). Em mecΓ’nica clΓ‘ssica, para partΓculas massivas, a energia depende do quadrado do momento ; para fΓ³tons a energia mostra-se diretamente proporcional ao momento por este transportado . Grandezas fΓsicas importantes sΓ£o definidas a partir da relação de dispersΓ£o apresentada por um dado ente, a exemplo a massa.
Como as transformaçáes observadas em um sistema tΓͺm naturezas as mais diversas, a exemplo indo desde uma simples mudanΓ§a nas velocidades das partΓculas do sistema atΓ© um rearranjo completo das posiçáes espaciais de partΓculas interagentes uma em relação Γ s outras e mesmo de um sistema inteiro em relação ao outro, para cada transformação define-se a forma de se determinar o valor da grandeza energia a ela associada, fazendo-se esta definição sempre de forma que as mudanΓ§as observadas neste caso sejam descritas por uma variação de energia igual em mΓ³dulo ao determinado para as variaçáes de energia associadas a todas as outras mudanΓ§as relacionadas, e de forma a garantir-se que a energia total dos sistemas em interação sempre se conserve.
Γ energia associada ao movimento dos corpos ou partΓculas dΓ‘-se o nome de energia cinΓ©tica, e mostra-se que esta deve ser determinada, em casos abrangidos pela fΓsica clΓ‘ssica, atravΓ©s da expressΓ£o: . Γ energia associada a entes fΓsicos mutuamente interagentes em virtude exclusiva das posiçáes espaciais que ocupam um em relação aos outros dΓ‘-se o nome de energia potencial. A forma de calculΓ‘-la Γ© determinada em acordo com a natureza da interação entre os mesmos. Quando a interação Γ©, a citar-se a interação entre o satΓ©lite e a terra como exemplo, de natureza gravitacional, a energia potencial associada recebe o nome de energia potencial gravitacional, e neste caso Γ© adequadamente calculada atravΓ©s da expressΓ£o:
, onde G Γ© a constante de gravitação universal, h a altura do satΓ©lite, RT o raio da Terra, m a massa do satΓ©lite e MT a massa da Terra. Repare a dependΓͺncia explicita da energia com a posição do satΓ©lite relativa Γ Terra, adequadamente representada pela distΓ’ncia (RT+h) do satΓ©lite ao centro do planeta, e com as massas da Terra e do satΓ©lite, refletindo o fato de tratar-se de uma interação de natureza gravitacional (onde massa atrai massa). Se a nature
|
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Wikipedia
| -0.103853 | 1.0804 | -1.031882 | 0.33494 |
za for elétrica, tem-se a energia potencial elétrica; se for elÑstica (lei de Hooke) tem-se a energia potencial elÑstica, e assim por diante, definindo-se sempre uma forma adequada de se calcular a energia associada de forma a ter-se sempre a lei da conservação da energia vÑlida, qualquer que sejam as naturezas das mudanças relacionadas ou os sistemas em interação.
No contexto de interação entre sistemas Γ© vital falar-se sobre uma entidade fΓsica amplamente encontrada ao abordar-se o assunto, principalmente quando o tema Γ© energia potencial: o campo. Inicialmente introduzido por Michael Faraday na FΓsica, este surge como uma mera simplificação matemΓ‘tica junto a solução de problemas prΓ‘ticos, mas com o avanΓ§o da tecnologia, verificou-se que o campo Γ© em verdade mais do que isto, vindo nos paradigmas modernos a ganhar o posto de ente fΓsico real. O fato empΓrico que leva Γ necessidade do conceito de campo mediando a interação entre sistemas Γ© o de que, para um observador externo aos sistemas que interagem, uma mudanΓ§a em um sistema nem sempre Γ© imediatamente acompanhada pela correspondente mudanΓ§a no outro sistema. HΓ‘ um lapso de tempo experimentalmente verificΓ‘vel e mensurΓ‘vel entre as duas mudanΓ§as que obriga a uma revisΓ£o do conceito de ação Γ distΓ’ncia que vigorou nas primeiras teorias sobre as interaçáes entre os entes fΓsicos, a exemplo na Gravitação universal de Newton. Se a energia liga-se diretamente Γ mudanΓ§as observadas no sistema, Γ© evidente que a energia do primeiro sistema diminui antes que a energia no segundo sistema aumente, o que em princΓpio violaria durante este lapso de tempo a lei da conservação da energia. Os resultados dos experimentos modernos demonstram entretanto que esta energia estΓ‘ literalmente a propagar-se pelo espaΓ§o entre os dois sistemas, estando esta associada ao campo fΓsico responsΓ‘vel pela interação entre eles. A velocidade na qual esta energia se propaga no vΓ‘cuo Γ© em verdade, qualquer que seja o referencial (inercial) adotado, a maior velocidade admissΓvel pela natureza para qualquer ente fΓsico, sendo essa conhecida na fΓsica pela letra c. Nos dias de hoje o valor dessa velocidade Γ© exatamente definido, valendo c = 299 792 458 metros por segundo, sendo as definiçáes de metro e segundo dela entΓ£o derivadas.
Em acordo com o paradigma moderno tem-se portanto que energia pura pode propagar-se pelo espaΓ§o na forma de um campo, existindo como um ente fΓsico real. Entre estes campos certamente o destaque Γ© para o campo eletromagnΓ©tico, que expressa a interação eletromagnΓ©tica entre partΓculas eletricamente carregadas. A esta energia pura propagando-se dΓ‘-se o nome de radiação eletromagnΓ©tica. A luz Γ© uma onda eletromagnΓ©tica, e como tal pode ser entendida como energia pura em movimento. Ao passo que a existΓͺncia das ondas eletromagnΓ©ticas encontra-se bem estabelecida, os cientistas ainda procuram observar ondas de campos associados Γ interaçáes de outras naturezas; a saber, no final de 2015, pesquisadores do projeto LIGO (Laser Interferometer Gravitational-Wave Observatory) observaram "distorçáes no espaΓ§o e no tempo" causadas por um par de buracos negros com 30 massas solares em processo de fusΓ£o.
O teorema de Noether
Uma profunda e abrangente consequΓͺncia da simetria presente na natureza encontra-se expressa em um teorema conhecido por Teorema de Noether. Em resumo, ele afirma que "toda simetria contΓnua no comportamento dinΓ’mico de um sistema - ou seja, na equação dinΓ’mica e no potencial mecΓ’nico - implica uma lei de conservação para aquele sistema. ... De enorme import
|
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Wikipedia
| 0.148281 | 1.179032 | -0.696017 | 0.303318 |
Γ’ncia para a termostΓ‘tica Γ© a simetria das leis da dinΓ’mica frente Γ translaçáes temporais. Isto Γ©, as leis fundamentais da dinΓ’mica (como as Leis de Newton, as equaçáes de Maxwell ou a Equação de SchrΓΆdinger) permanecem inalteradas mediante a transformação t --> t' + t0 (ou seja, por uma mudanΓ§a na origem da escala de tempo). Se o potencial externo Γ© independente do tempo, o teorema de Noether prediz a existΓͺncia de uma quantidade que se conserva. Esta quantidade Γ© nomeada energia.".
Unidades
A unidade de energia no sistema internacional de unidades Γ© o joule (J). O joule Γ© uma unidade derivada, equivalente a 1 newton metro () ou ainda a 1 quilograma metro quadrado por segundo quadrado ().
1 joule corresponde à energia transferida a um objeto por uma força resultante constante de 1 N que, atuando de forma sempre paralela à trajetória descrita, o faz durante o intervalo de tempo necessÑrio para que este objeto mova-se 1 metro ao longo da trajetória.
Embora a unidade oficial seja o joule, outras unidades de energia sΓ£o frequentemente utilizadas em função do contexto. Destacam-se o (quilo)watt-hora (kWh) , unidade utilizada na medida do consumo de energia elΓ©trica residencial ou industrial, o elΓ©tron-volt (eV), muito utilizada em fΓsica nuclear e de fΓsica de partΓculas, e o erg, unidade muito comum em paΓses que ainda nΓ£o adotaram por completo o estabelecido pelo Sistema Internacional de Unidades.
O watt-hora corresponde Γ energia transformada quando um dispositivo cuja potencia seja de 1 watt opera durante um intervalo de tempo de 1 hora. Uma lΓ’mpada cuja potΓͺncia nominal Γ© 60Β W transforma 720Β Wh (ou seja, 0,72Β kWh) de energia elΓ©trica em outras formas de energia a cada 12 horas de funcionamento (720Β Wh = 60Β W x 12Β h).
O elétron-volt corresponde à energia cinética ganha quando um elétron move-se entre dois pontos separados por uma diferença de potencial de 1 volt.
O erg Γ© a unidade utilizada ao empregar-se o sistema de unidade cgs, comum em alguns paΓses mesmo hoje em dia. Um erg equivale a um grama centΓmetro quadrado por segundo quadrado, ou seja, Γ dΓ©cima milionΓ©sima parte do joule (1 erg = 10β7 joules).
Formas de energia
Apesar de nΓ£o se restringir a isso, a energia pode ser entendida como a capacidade de realizar trabalho, a capacidade de colocar as coisas em movimento, e movimento Γ© algo fundamental no nosso dia-a-dia. As sociedades humanas dependem cada vez mais de um elevado consumo energΓ©tico para sua subsistΓͺncia. Para isso foram sendo desenvolvidos ao longo da histΓ³ria diversos processos de transformação, transporte e armazenamento de energia. Na realidade, em acordo com o expresso pela primeira lei da termodinΓ’mica e pelos conceitos de energia interna e energia tΓ©rmica, sΓ³ existem, alΓ©m da energia pura radiante, duas formas de energia armazenadas em um sistema: a potencial e a cinΓ©tica. No cotidiano entretanto estas acabam recebendo nomes especΓficos que geralmente fazem referΓͺncia explΓcita Γ natureza do sistema envolvido no armazenamento ou Γ s plantas industriais onde estas sΓ£o levadas Γ transformação. Assim tem-se a energia hidrΓ‘ulica como sinΓ΄nimo de energia potencial gravitacional ou mesmo cinΓ©tica armazenada nas Γ‘guas de uma represa hidroelΓ©trica, que conforme o nome diz, cuida da conversΓ£o de energia "hidrΓ‘ulica" em energia potencial elΓ©trica; a energia nuclear para a
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Wikipedia
| -0.3939 | 0.491206 | -0.470827 | 0.357707 |
energia potencial associada à interação nuclear forte, ou até mesmo, em senso comum, para a energia elétrica produzida em termoelétricas cujas fontes de energia térmica sejam reatores nucleares; a energia eólica associada à energia cinética de movimento das massas de ar (ventos); a energia solar associada à radiação eletromagnética com origem no Sol e energia geotérmica associada à energia térmica do interior da terra.
Energia potencial
A energia potencial é aquela associada à posição. Um martelo levantado, uma mola comprimida ou esticada ou um arco tensionado de um atirador, todos possuem energia potencial. Esta energia estÑ pronta para ser transformada em outras formas de energia e serÑ transformada, mediante a realização de trabalho, tão logo a configuração espacial do sistema que contém a energia potencial mude: quando o martelo cair, pregarÑ um prego; a mola, quando solta, farÑ andar os ponteiros de um relógio; o arco dispararÑ uma flecha. Assim que ocorrer algum movimento, a energia potencial da fonte diminui, enquanto se transforma nos casos citados em energia de movimento (energia cinética). Ao contrÑrio, levantar o martelo, comprimir a mola e esticar o arco são processos onde a energia cinética transforma-se em energia potencial.
Normalmente atribui-se a energia potencial ao objeto que ocupa uma dada posição dentro do sistema ao qual pertence, como feito anteriormente. Ressalva-se explicitamente entretanto que a energia nΓ£o pertence exclusivamente ao objeto como parece Γ primeira vista. Esta encontra-se em verdade armazenada no sistema como um todo, composto pelo objeto e suas demais partes. Muitas vezes nΓ£o faz-se referΓͺncia explΓcita ao resto do sistema, mas este sempre figura, se nΓ£o de forma explicita, pelo menos adequadamente substituΓdo por um campo bem determinado, que responde pela interação do objeto com o sistema em questΓ£o, mesmo que o faΓ§a de forma implΓcita. Fala-se assim da energia potencial gravitacional de um aviΓ£o - no campo de gravidade da Terra -, de energia potencial de um elΓ©tron - no campo elΓ©trico gerado pelos polos de uma bateria -, e assim por diante.
Uma consideração importante sobre a energia potencial refere-se Γ sua medida. NΓ£o se determina fisicamente o valor absoluto da energia potencial de um sistema em uma dada configuração, mesmo porque isto nΓ£o faria muito sentido. O que Γ© fisicamente mensurΓ‘vel Γ© a variação da energia potencial observada quando o sistema muda sua configuração, indo de um estado inicial para um estado final. Nestes termos Γ© usual atribuir-se uma energia potencial nula (zero) para o sistema em uma dada configuração espacial inicialmente especificada, e entΓ£o medir-se a energia potencial de qualquer outra configuração do sistema em relação a este estado de referΓͺncia, sendo a energia potencial de uma configuração qualquer igual Γ energia que teve que ser transferida ao sistema para levΓ‘-lo do estado de referΓͺncia atΓ© esta configuração final, mantidas as energias cinΓ©ticas associadas Γ s partes integrantes do sistema constantes de forma que toda a energia entregue ao sistema seja inteiramente armazenada na forma de energia potencial.
A energia potencial Γ© assim dependente de um referencial a se escolher no inΓcio do problema - e que deve ser mantido durante todo o problema sobre risco de obter-se uma solução incorreta. A energia potencial de uma lΓ’mpada em relação ao piso de um apartamento de cobertura Γ© certamente diferente da energia potencial da mesma lΓ’mpada se a referΓͺncia adotada for o solo, em nΓvel do andar tΓ©rreo.
No cotidiano encontram-se presentes diversos tipos de energia potencial,
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] | 1,024 |
Wikipedia
| -0.018266 | 0.732594 | -1.026503 | 0.157857 |
dos quais se destacam: a elΓ‘stica, a gravitacional e a elΓ©trica.
Energia potencial gravitacional
A energia potencial gravitacional entre duas massas passΓveis de serem tratadas como massas pontuais Γ© fornecida pela Teoria da gravitação universal, sendo expressa pela relação:
,
onde m1 e m2 sΓ£o as respectivas massas das partΓculas, r a distΓ’ncia entre elas, e G a Constante gravitacional universal (cuja função Γ© estabelecer as unidades a se usarem na expressΓ£o). Nesta expressΓ£o o sistema de referΓͺncia para o qual a energia potencial Γ© definida como nula Γ© aquele composto pelas massas infinitamente afastadas. Como a forΓ§a de gravidade Γ© sempre atrativa, a energia potencial para duas massas juntas Γ© sempre menor do que para as mesmas massas separadas: a energia potencial Γ©, assim, negativa para qualquer par de massas separadas por uma distΓ’ncia mensurΓ‘vel (nΓ£o infinita).
Isaac Newton demonstrou de forma muito elegante, atravΓ©s do desenvolvimento do cΓ‘lculo integral e diferencial, que para interaçáes como a gravitacional e a elΓ©trica - que dependem do inverso do quadrado da distΓ’ncia - distribuiçáes esfericamente simΓ©tricas e homogΓͺneas de massa ou carga podem ser, para todos os efeitos externos Γ estas, consideradas como se fossem partΓculas pontuais situadas nos centros das esferas, sendo a massa ou a carga destas partΓculas iguais Γ massa ou carga totais presentes nestas esferas. Dai o uso do raio da Terra para calcular-se o campo gravitacional em sua superfΓcie. Pelo mesmo motivo a Terra pode ser considerada um excelente terra elΓ©trico. Tal comportamento tambΓ©m Γ© facilmente demonstrado atravΓ©s da aplicação da Lei de Gauss aos sistemas em questΓ£o, sendo conhecido por "teorema das cascas".
A energia potencial de interação entre dois objetos quaisquer do dia-a-dia Γ©, em virtude dos pequenos valores das duas massas envolvidas, muito pequena, sendo desprezΓvel para qualquer problema prΓ‘tico. A energia potencial gravitacional Γ© particularmente importante quando um objeto Γ© muito massivo: a Terra por exemplo. A energia potencial gravitacional de um objeto nas proximidades da superfΓcie da Terra Γ© proporcional Γ altura (h) deste corpo - medida, conforme jΓ‘ exposto, em relação a um dado nΓvel de referΓͺncia previamente escolhido para o qual atribui-se uma energia potencial zero, sendo este agora o nΓvel do solo no local em questΓ£o e nΓ£o o infinito, como no caso anterior. Nestes termos a energia potencial de um objeto pode ser calculada pela expressΓ£o:
,
onde p Γ© o peso do objeto, P = m. g, donde:
Repare que, embora grandezas relativas Γ Terra nΓ£o apareΓ§am explicitamente nesta expressΓ£o, a energia potencial encontra-se necessariamente associada ao sistema Terra objeto e nΓ£o apenas ao objeto; a Terra encontra-se representada neste caso pelo valor do campo de gravidade g existente junto Γ superfΓcie do planeta e determinado segundo a gravitação universal por:
CΓ‘lculos feitos, tem-se para para o campo junto Γ superfΓcie da terra o valor aproximado de 9,8 metros por segundo quadrado.
A energia potencial assim determinada serΓ‘ positiva para o objeto em pontos acima do nΓvel de referΓͺncia (altura positiva) e negativas para o objeto situado em pontos abaixo deste nΓvel (altura negativa).
A expressão Epg=mgh vale apenas para pequenas alturas se comparadas ao raio RT da Terra, onde o campo pode ser considerado constante. Para alturas considerÑveis define-se a energia potencial nula para a configuração em que o objeto e o planeta encontram-se infinitamente distantes, e, neste caso, a energia potencial de uma sistema é, novamente com o referen
|
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] | 1,024 |
Wikipedia
| -0.183327 | 0.997342 | -0.693579 | 0.205922 |
cial no infinito:
Repare que embora o valor absoluto da energia potencial seja muito dependente do sistema adotado como referΓͺncia - para o qual a energia potencial Γ© definida como zero -, a variação da energia potencial ocorrida quando o sistema muda sua configuração espacial, indo de um estado inicial para um final, serΓ‘ sempre a mesma, qualquer que seja o sistema de referΓͺncia adotado.
A variação na energia potencial gravitacional calculada segundo a última expressão coincide (em primeira ordem) com a calculada através da expressão para pequenas variaçáes de altura, ou seja, para pequeno.
Energia potencial elΓ©trica
Para interaçáes entre partΓculas pontuais a energia potencial elΓ©trica Γ© a energia associada a uma partΓcula qualquer com carga elΓ©trica "q" situada a uma distΓ’ncia "d" de uma outra partΓcula com carga "Q". Γ calculada pela expressΓ£o:
Nesta expressΓ£o a configuração para a energia potencial nula Γ© aquela onde as cargas encontram-se infinitamente distantes umas das outras. Se as cargas tΓͺm mesmo sinal e se repelem, o sistema por elas formado quando encontram-se separadas por uma distΓ’ncia r nΓ£o infinita tem energia potencial positiva. No caso em que as cargas tΓͺm sinais contrΓ‘rios hΓ‘ uma atração entre as mesmas, e na formação do sistema a partir das mesmas no infinito deve-se remover energia do sistema no processo a fim de ter-se as cargas estΓ‘ticas; a energia potencial do sistema formado serΓ‘ negativa.
Tem-se da teoria do eletromagnetismo que o potencial elΓ©trico V de um ponto situado a uma distΓ’ncia d de uma carga Q Γ© dado por:
,
donde:
A ΓΊltima expressΓ£o tem em verdade validade geral, nΓ£o sendo exclusiva para casos envolvendo duas cargas pontuais. Γ muito ΓΊtil em anΓ‘lise de circuitos, e o potencial de referΓͺncia (zero volt) nΓ£o precisa estar no infinito, podendo neste caso ser um ponto de referΓͺncia escolhido livremente dentro do circuito. O cΓ‘lculo do potencial do ponto entretanto nΓ£o Γ© mais dado pela expressΓ£o que a antecede visto que nΓ£o hΓ‘ claramente neste caso apenas uma carga pontual responsΓ‘vel pelo potencial no referido ponto.
Tem-se respetivamente, nas expressΓ΅es:
= constante eletrostΓ‘tica do meio em que as cargas estiverem inseridas;
= potencial elétrico do ponto onde coloca-se a carga q devido à presença da carga Q ou de qualquer outro sistema de cargas;
= carga da partΓcula Γ qual "associa-se" a energia potencial elΓ©trica, tambΓ©m chamada carga de prova;
= distΓ’ncia entre a carga q (pontual) e a carga fonte Q (tambΓ©m pontual);
= carga fonte Q (pontual).
Energia potencial elΓ‘stica
A energia potencial elΓ‘stica estΓ‘ associada a uma mola ou a um corpo deformado desde que em regime elΓ‘stico e nΓ£o plΓ‘stico. Em detalhes, em termos de estrutura da matΓ©ria, a energia potencial elΓ‘stica relaciona-se diretamente Γ s energias potenciais elΓ©trica existente entre as partΓculas que compΓ΅em o corpo, possuindo ambas, em essΓͺncia, a mesma natureza.
Γ calculada pela expressΓ£o (mola ideal):
,
onde:
K = a constante elΓ‘stica da mola, a mesma dada estabelecida pela lei de Hooke (em newtons por metro).;
X = a elongação, a variação no tamanho da mola (em metros).
Esta expressão assume a configuração de energia potencial nula a configuração para a mola solta, em seu tamanho natural. Como a elongação aparece quadrada
|
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] | 1,024 |
Wikipedia
| -0.248877 | 1.018107 | -0.926082 | 0.243869 |
, tanto faz esticar como comprimir a mola, a energia associada serÑ sempre positiva. As variaçáes nesta energia podem perfeitamente ser negativas, entretanto.
Energia potencial nuclear
Convém abrir-se esta seção com algumas consideraçáes importantes apresentadas por Robert Eisberg em um famoso livro didÑtico de sua autoria:
" Apesar de dispormos atualmente de um conjunto bastante completo sobre as forΓ§as nucleares, contata-se que elas sΓ£o demasiadamente complicadas, nΓ£o sendo possΓvel atΓ© agora usar este conhecimento para produzir uma teoria ampla dos nΓΊcleos. Em outras palavras, nΓ³s nΓ£o podemos explicar todas as propriedades dos nΓΊcleos em função das propriedades das forΓ§as nucleares que atuam sobre seus prΓ³tons e nΓͺutrons. Existem entretanto diversos modelos ... Cada um deles pode explicar um certo nΓΊmero limitado de propriedades nucleares ..."
Ainda encontra-se no mesmo livro: " Uma diferenΓ§a profunda entre o estudo experimental dos nΓΊcleos e dos Γ‘tomos decorre da diferenΓ§a entre suas energias caracterΓsticas. A energia caracterΓstica dos nΓΊcleos Γ© da ordem de 1 Mev... Veremos um pouco mais Γ frente que esta mesma ordem de grandeza caracteriza a energia de ligação de um prΓ³ton ou nΓͺutron em um nΓΊcleo tΓpico assim como a energia de separação entre seu estado fundamental e o primeiro estado excitado. A energia caracterΓstica dos Γ‘tomos Γ© da ordem de 1 eV." , mil vezes menor, portanto.
Ressalvas acima consideradas, define-se energia nuclear como a energia potencial associada Γ posição relativa dos nucleΓ΄ns um em relação aos outros em virtude da interação nuclear forte que os mantΓ©m unidos no nΓΊcleo atΓ΄mico, definição razoΓ‘vel ao se considerar os modelos para os nΓΊcleos propostos, a citar: o modelo nuclear da gota lΓquida, o modelo do gΓ‘s de fermi, o modelo de camadas, o modelo coletivo, e outros.
A forΓ§a nuclear forte, ao contrΓ‘rio da elΓ©trica e da gravitacional, apesar de atrativa Γ© uma forΓ§a de curto alcance: possui um valor extremamente alto se comparado Γ elΓ©trica quando dois nucleΓ΄ns estΓ£o a uma distΓ’ncia curta e decai rapidamente a zero se estes se afastam alΓ©m de uma certa distΓ’ncia limite. "ela atua de maneira apreciΓ‘vel somente em uma distΓ’ncia inferior a 10Β F" (1Β F = 1Β fermi = 10β15Β m, aproximadamente o raio de um prΓ³ton ou nΓͺutron). Considerando-se o sistema com os nucleΓ΄ns "infinitamente" separados como referΓͺncia para a medida da energia potencial nuclear (zero neste caso), isto traduz-se em uma energia potencial negativa muito elevada para o nΓΊcleo formado. A energia potencial nuclear negativa confina os prΓ³tons e nΓͺutrons no interior do nΓΊcleo mesmo sob a intensa repulsΓ£o elΓ©trica experimentada pelos prΓ³tons devido Γ sua proximidade pois, neste Γ’mbito, a energia potencial nuclear Γ©, em mΓ³dulo, muito superior Γ energia potencial elΓ©trica - positiva - associada aos nucleΓ΄ns carregados. A energia potencial elΓ©trica liberada caso um prΓ³ton venha a escapar do nΓΊcleo sob a ação da forΓ§a elΓ©trica nΓ£o Γ© capaz de compensar o aumento na energia potencial nuclear associado a esta fuga, isto em situaçáes comuns, pelo menos. "ExperiΓͺncias recentes envolvendo espalhamento de prΓ³tons por prΓ³tons mostra que o alcance das forΓ§as nucleares Γ© da ordem de 2Β F e que o valor de energia associada Γ forΓ§a atrativa Γ© aproximadamente 10 vezes maior do que a energia coulombiana quando os dois prΓ³tons se encontram separados por esta distΓ’ncia".
Variaçáes nas energias potenciais nucleares ocorrem quando o núcleo participa de uma reação nuclear. As energias
|
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Wikipedia
| -0.3939 | 0.737785 | -0.606258 | -0.136859 |
liberadas neste processo sΓ£o ordens de grandeza maiores do que as liberadas a partir de variaçáes nas energias quΓmicas associadas Γ eletrosfera deste Γ‘tomo quando este participa de uma reação quΓmica.
Energia cinΓ©tica
Γ a energia que um corpo massivo em movimento possui devido Γ sua velocidade. Uma questΓ£o importante a levantar-se aqui Γ© que a energia cinΓ©tica Γ©, em virtude da relatividade do movimento, fortemente dependente do referencial adotado para seu cΓ‘lculo. Para um observador fixo ao solo, o motorista de um Γ΄nibus em movimento - assumido um movimento uniforme por simplicidade - estΓ‘ animado com uma velocidade , e por tal encontra-se dotado com uma energia cinΓ©tica nΓ£o nula. Contudo, para um passageiro sentado no banco do mesmo Γ΄nibus, o mesmo motorista nΓ£o encontra-se animado, e sendo sua velocidade relativa a este referencial nula, sua energia cinΓ©tica tambΓ©m deve sΓͺ-lo. Para o passageiro no banco do Γ΄nibus Γ© o observador no solo que encontra-se dotado com energia cinΓ©tica, e nΓ£o o motorista. Contudo, ao contrΓ‘rio do que a primeira impressΓ£o possa sugerir, nΓ£o hΓ‘, em vista do princΓpio da conservação da energia, necessΓ‘ria correspondΓͺncia entre os valores destas energias, justamente por terem sido medidas em diferentes referenciais.
A conservação da energia sempre Γ© observada em um mesmo referencial, qualquer que seja o referencial inercial escolhido, contudo seus valores absolutos sΓ£o altamente dependentes do referencial escolhido, e a lei da conservação da energia nΓ£o implica que estes valores sejam diretamente compatΓveis com as mudanΓ§as de referencial que por ventura venham a se realizar durante a solução do problema em consideração.
A expressΓ£o para calcular-se a energia cinΓ©tica mostra-se tambΓ©m dependente do escopo em consideração, sendo relativamente simples na mecΓ’nica clΓ‘ssica e um pouco mais complicada no Γ’mbito da relatividade restrita ou teorias mais avanΓ§adas. Em mecΓ’nica clΓ‘ssica hΓ‘ a energia cinΓ©tica translacional, associada Γ translação de uma partΓcula ou do centro de massa de um sistema, e a energia cinΓ©tica rotacional, associada Γ rotação de um corpo rΓgido em torno de um eixo de rotação que passe por seu centro de massa. Contudo, antes de entrar-se diretamente em consideraçáes quantitativas sobre estas, Γ© valido falar-se um pouco sobre uma forma de energia cinΓ©tica que nΓ£o encontra-se diretamente associada Γ translação do centro de massa de um sistema ou rotação em torno deste centro, mas sim presa dentro de um sistema na forma de energia cinΓ©tica associada Γ agitação tΓ©rmica das partΓculas que o integram: a energia tΓ©rmica.
Energia tΓ©rmica
A energia tΓ©rmica Γ©, no fundo, energia cinΓ©tica. A distinção entre "energia tΓ©rmica" e "energia cinΓ©tica" Γ© necessΓ‘ria apenas em virtude de escala. Para sistemas encarados explicitamente a partir de cada uma das partΓculas que o compΓ΅em, partΓculas aqui em acepção de constituintes os mais bΓ‘sicos da matΓ©ria, sΓ³ hΓ‘ energia cinΓ©tica, a saber a translacional, explicitamente determinada para cada partΓcula. Nesta escala e apenas nesta escala "energia" Γ© aceitavelmente definida como a capacidade de produzir trabalho. Entretanto, para sistemas (corpos) macroscΓ³picos compostos por um gigantesco amontoado destas agora "invisΓveis" partΓculas - os estudados pela termodinΓ’mica - Γ© conveniente e em verdade necessΓ‘rio distinguir entre a parcela de energia cinΓ©tica total das partΓculas microscΓ³picas nΓ£o associada Γ translação do sistema - a chamada energia tΓ©rmica (microscΓ³pica), esta nΓ£o diretamente perceptΓvel em escala macroscΓ³pica - e a parcela desta energia que encontra-se
|
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] | 1,024 |
Wikipedia
| -0.265519 | 1.0804 | -1.265001 | 0.050342 |
associada Γ translação ou mesmo rotação do sistema como um todo, ou seja, Γ translação do centro de massa do sistema ou rotação do sistema em torno deste, esta diretamente perceptΓvel em escala macroscΓ³pica. Estas ΓΊltimas sΓ£o a energia cinΓ©tica de translação e rotação conforme abaixo definidas para os corpos clΓ‘ssicos (ou para os "imaginados" como macroscΓ³picos).
Em termodinΓ’mica a transferΓͺncia de energia cinΓ©tica ou a sua conversΓ£o em energia potencial ou de potencial nesta implicam visivelmente em trabalho: qualquer variação de energia cinΓ©tica (doravante sempre macroscΓ³pica) sempre implica trabalho; a transformação de energia potencial ou cinΓ©tica (de energia mecΓ’nica) em tΓ©rmica tambΓ©m Γ© feita a princΓpio mediante trabalho (doravante sempre macroscΓ³pico), mas este trabalho, ao aumentar a energia tΓ©rmica do sistema, implica sua "conversΓ£o" imediata em calor, sendo o calor uma resultante direta da transferΓͺncia de energia tΓ©rmica dentro do sistema ou mesmo entre este e outros sistemas vizinhos que ocorre em virtude da diferenΓ§a de temperaturas estabelecida pelo acrΓ©scimo de energia tΓ©rmica no dado ponto do sistema envolvido no trabalho em questΓ£o (em palavras mais simples, o atrito "aquece"). Calor, na prΓ‘tica, implica sempre em aumento da entropia, o que literalmente implica que parte da energia cinΓ©tica inicial que fora transformada em energia tΓ©rmica mediante este trabalho, uma vez integrado Γ energia interna do sistema, torna-se permanentemente indisponΓvel Γ realização de qualquer outro trabalho, nunca mais "reaparecendo" em forma de energia cinΓ©tica no mundo macroscΓ³pico. A parcela de energia tΓ©rmica associada ao aumento da entropia Γ© literalmente e definitivamente "perdida" para as "entranhas" do sistema. Mesmo em uma mΓ‘quina tΓ©rmica - especialmente projetada para fazer a transformação inversa, realizar trabalho Γ s expensas de calor - esta parcela de energia nΓ£o poderΓ‘ mais ser convertida em energia cinΓ©tica mensurΓ‘vel; mas ela ainda encontra-se lΓ‘, presa dentro do sistema (e "mensurΓ‘vel" em uma escala microscΓ³pica).
Nesta escala, onde valem as leis da termodinΓ’mica, definir "energia" como a capacidade de realizar trabalho mostra-se "delicado" de ser feito, portanto.
Energia cinΓ©tica translacional
Retomando-se aos casos associados ao centro de massa - quer macroscΓ³picos que no caso de uma partΓcula - a energia cinΓ©tica Γ© calculada no Γ’mbito da fΓsica clΓ‘ssica, para o caso translacional, por:
,
onde:
= massa do corpo.
= velocidade do centro de massa do corpo.
Resolvendo-se o produto escalar, em termos do mΓ³dulo da velocidade , esta expressΓ£o traduz-se por:
Isto significa que quanto mais rΓ‘pido um dado objeto se move maior Γ© a quantidade de energia cinΓ©tica que o mesmo possui. AlΓ©m disso, quanto mais massivo for o objeto, maior serΓ‘ a quantidade de energia cinΓ©tica presente quando este estiver se movendo a uma dada velocidade.
Para uma partΓcula pontual, mesmo microscΓ³pica, se a velocidade em consideração for a velocidade desta em relação Γ origem do referencial adotado, o que geralmente o Γ©, a expressΓ£o acima representa a energia cinΓ©tica total que esta possui. Entretanto, para corpos extensos (com dimensΓ΅es), alΓ©m de transladar este pode girar, e a energia cinΓ©tica conforme calculada acima constitui-se apenas em uma parcela da sua energia cinΓ©tica macroscΓ³pica total.
Para que algo se mova Γ© necessΓ‘rio transformar qualquer outro tipo de energia em energia cinΓ©tica. As mΓ‘quinas mecΓ’nicas - automΓ³veis, torno mecΓ’nico, b
|
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Wikipedia
| -0.248282 | 0.971387 | -1.304452 | -0.096383 |
ate-estacas ou quaisquer outras mΓ‘quinas motorizadas - transformam algum tipo de energia, geralmente previamente armazenada na forma de alguma energia potencial, em energia cinΓ©tica.
Para variar-se a energia cinética total de um objeto necessita-se realizar sobre o mesmo um trabalho. Isto traz à luz o teorema do trabalho - variação da energia cinética, que afirma a igualdade entre os valores do trabalho realizado e a variação da energia cinética apresentada pelo corpo.
Relembrando mais uma vez, vale ressaltar que a energia cinΓ©tica, assim como a energia potencial, nΓ£o Γ© absoluta. A energia cinΓ©tica de um corpo Γ© dependente do referencial adotado para fazer-se a medida da velocidade deste corpo. Isto decorre diretamente da relatividade do movimento.
No Γ’mbito de outras teorias para a dinΓ’mica mais abrangentes, a energia cinΓ©tica pode ser definida por uma expressΓ£o bem diferente da encontrada no escopo da mecΓ’nica clΓ‘ssica. A exemplo, a energia cinΓ©tica de uma partΓcula com massa de repouso m0 que se move com uma velocidade v Γ© definida, no Γ’mbito da relatividade especial, por:
Esta expressΓ£o se reduz Γ apresentada para o caso da mecΓ’nica clΓ‘ssica quando a velocidade v do objeto Γ© muito inferior Γ velocidade da luz c, conforme esperado.
O autor Γ© remetido ao estudo das respectivas teorias para maiores detalhes, se necessΓ‘rio.
Energia cinΓ©tica rotacional
A chamada energia rotacional Γ© simplesmente a energia cinΓ©tica associada a um corpo material extenso (ou nΓ£o) que executa um movimento de rotação em torno de um eixo de referΓͺncia que pode ou nΓ£o atravessΓ‘-lo, sem que este entretanto translade (o eixo Γ© fixo no referencial adotado, e passa pois pelo centro de massa do corpo). Γ determinada a partir da soma - da integral - da energia cinΓ©tica que cada pedacinho de massa em que se pode dividi-lo tem devido Γ rotação, sendo esta integral feita ao longo de todo o corpo. Repare que um pedacinho do corpo, quando prΓ³ximo ao eixo de rotação, tem energia cinΓ©tica menor pois move-se tambΓ©m com velocidade tangencial menor se comparado a um pedacinho similar que encontre-se situado longe do eixo de rotação. Em termos de mecΓ’nica rotacional, esta integral, ao ser realiza, resulta em:
onde I representa o momento de inércia deste corpo em relação ao eixo em questão e representa a velocidade angular do corpo em relação ao mesmo eixo.
Ao passo que para variar-se a energia cinética de translação necessitamos de uma força que realize um trabalho, para variar-se a energia de rotação esta força deve também prover um torque, e através dele também realizar trabalho.
Energia cinΓ©tica total
A energia cinΓ©tica total de um corpo rΓgido que alΓ©m de rotacionar tambΓ©m translada, a exemplo uma esfera que rola sobre um plano inclinado sem escorregar, ou mesmo uma roda de bicicleta movendo-se em contato com o solo, Γ© dada pela sua energia cinΓ©tica de rotação em torno do eixo de rotação mais a energia cinΓ©tica a ele associada devido Γ translação deste eixo:
onde m representa a massa total do corpo, v a velocidade de translação do centro de massa do sistema, a velocidade angular do sistema em torno do eixo de rotação - que passa pelo centro de massa do sistema - e I o momento de inércia do corpo em torno do eixo em consideração.
O teorema do trabalho - variação da energia cinética aplica-se à energia total de um corpo.
Cargas elΓ©tricas em movimento
Quando cargas elΓ©tricas sΓ£o colocadas
|
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Wikipedia
| -0.446203 | 0.820843 | -0.960153 | 0.428541 |
em movimento de forma a estabelecer uma corrente elΓ©trica, esta produz ao seu redor um campo magnΓ©tico. Correntes constantes mantΓ©m o campo constante, e hΓ‘ uma energia associada a este campo, podendo esta ser chamada de energia magnΓ©tica. A energia magnΓ©tica nΓ£o pode ser descrita atravΓ©s de uma "energia potencial magnΓ©tica" conforme ocorre para o caso da energia elΓ©trica porque o campo magnΓ©tico nΓ£o Γ© um campo conservativo. Mesmo o processo de variação da energia magnΓ©tica envolve um processo elΓ©trico - o princΓpio da indução eletromagnΓ©tica -, nΓ£o havendo mecanismos unicamente magnΓ©ticos capazes de descrevΓͺ-lo.
Conclui-se que uma partΓcula carregada em movimento possui uma quantidade de energia extra armazenada no campo magnΓ©tico e nΓ£o apenas a energia cinΓ©tica associada Γ sua massa em movimento.
O leitor Γ© remetido ao estudo da magnetostΓ‘tica e do eletromagnetismo para maiores detalhes.
Energia mecΓ’nica
No Γ’mbito da mecΓ’nica clΓ‘ssica, a energia mecΓ’nica de um sistema discreto de partΓculas ou corpos extensos Γ© a soma de todas as energias potenciais associadas Γ s interaçáes conservativas entre os corpos ou partΓculas em consideração, e de todas as energias cinΓ©ticas destes corpos ou partΓculas, incluΓdas as energias cinΓ©ticas de rotação, se aplicΓ‘vel.
A energia mecΓ’nica Γ©, em princΓpio, uma energia definida em escopo macroscΓ³pico - ou seja, para um sistema de corpos extensos - sendo o resultado da soma das energias cinΓ©ticas de translação dos centros de massa das partes do sistema, das energia cinΓ©ticas de rotação destas partes em torno dos respectivos centros de massa, e das energias potenciais devidas Γ interaçáes conservativas - como a de origem gravitacional, elΓ‘stica, ou elΓ©trica - entre essas partes. Em sistemas macroscΓ³picos, a energia tΓ©rmica, a energia quΓmica e outras parcelas associadas Γ s energias internas das partes nΓ£o integram, pois, a energia mecΓ’nica do sistema.
Contudo, no Γ’mbito da fΓsica estatΓstica, ao se estudarem os sistemas termodinΓ’micos - a saber, a matΓ©ria - o conceito de energia mecΓ’nica, quando aplicado microscopicamente Γ s partΓculas fundamentais que constituem um corpo material - suposto macroscopicamente estΓ‘tico no referencial adotado - leva diretamente ao conceito de energia interna de um sistema, correspondendo esta Γ soma de duas parcelas: a energia tΓ©rmica - atrelada diretamente Γ soma das energias cinΓ©ticas das partΓculas em escala microscΓ³pica e Γ temperatura absoluta do sistema - e a energia quΓmica, parcela correspondente Γ soma da(s) energia(s) potencial(is) devidas Γ s interaçáes - neste caso sempre conservativas - entre as partΓculas do sistema, a destacar-se de longe nessa escala a interação elΓ©trica entre elΓ©trons e nΓΊcleos, entre Γ‘tomos, entre molΓ©culas, etc..
A energia mecΓ’nica "EM" que um ΓΊnico corpo possui Γ© a soma da sua energia cinΓ©tica "Ec" com a(s) energia(s) potencial(is) Γ (s) qual(is) se sujeita em virtude de campos externos.
Se o sistema for conservativo, ou seja, apenas forças conservativas atuam sobre ele, a energia mecÒnica total se conserva e é uma constante de movimento.
O atrito nΓ£o Γ© uma forΓ§a conservativa. Sistema sujeitos a atrito tΓͺm sua energia mecΓ’nica afetada pelo mesmo.
Massa
Com o desenvolvimento da fΓsica moderna verificou-se, a partir dos resultados oriundos tanto da fΓsica quΓ’ntica quanto da fΓsica relativΓstica, que massa e energia sΓ£o intercambiΓ‘veis
|
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Wikipedia
| -0.096466 | 1.230944 | -0.730621 | 0.519612 |
, podendo ser convertidas uma na outra mediante processos fΓsicos hoje bem-estabelecidos. A equivalΓͺncia entre energia e massa Γ© expressa atravΓ©s da mundialmente conhecida equação E=mc2, proposta por Einstein ainda quando da publicação da relatividade especial.
A conversΓ£o de massa em energia encontra-se diretamente ligada Γ energia nuclear, pois em reaçáes nucleares altamente exoenergΓ©ticas, como a fissΓ£o do urΓ’nio ou a fusΓ£o do hidrogΓͺnio, verifica-se que a soma das massas dos produtos formados Γ© menor do que a soma das massas dos reagentes, sendo a diferenΓ§a inteiramente convertida em energia e liberada no processo. Processo que envolvem a criação de pares, como o que dΓ‘ origem a um pΓ³sitron e a um elΓ©tron a partir de energia pura (energia radiante), ou a aniquilação destes, com a liberação da energia associada, sΓ£o muito comuns em fΓsica de partΓculas.
Fatos experimentais que explicitam a conversΓ£o de massa em energia e energia em massa como processos naturais trazem Γ tona um problema com duas leis de conservação encontradas no Γ’mbito da mecΓ’nica clΓ‘ssica de formas completamente separadas: a lei da conservação de massas e a lei da conservação da energia (em sua forma clΓ‘ssica). Certamente a conversΓ£o entre massa em energia leva Γ violação de tais leis. Contudo ressalta-se que no mundo clΓ‘ssico, aquele acessΓvel aos nossos sentidos, no qual nos preocupamos com as reaçáes quΓmicas mas nΓ£o com as nucleares, a quantidade de massa que converte-se em energia ou vice-versa Γ© imperceptΓvel aos melhores equipamentos: no mundo clΓ‘ssico massa e energia se conservam de fora separada. Em fΓsica de altas energias, contudo, nΓ£o hΓ‘ lei de conservação de massa. HΓ‘ apenas lei da conservação da energia em sua forma abrangente, e a massa figura nesta lei mediante a famosa equação de Einstein, sendo tratada como uma forma de energia. A relação entre massa e energia encontra-se evidente na relatividade aos considerarmos a expressΓ£o: "A energia tem inΓ©rcia". Decorre que ao aumentar-se a energia de um sistema, aumenta-se tambΓ©m a sua inΓ©rcia ao responder a forΓ§as aplicadas, ou seja, a sua massa. Repare que nΓ£o hΓ‘ a necessidade explΓcita de conversΓ£o de energia em massa de repouso, e dizer que a massa aumentou nΓ£o significa necessariamente que matΓ©ria surgiu dentro do sistema. HΓ‘ assim uma clara distinção entre massa e massa de repouso. A massa de repouso de uma partΓcula em velocidade prΓ³xima Γ da luz, digamos, a de um elΓ©tron, continua a mesma, mas ao se tentar aumentar a velocidade deste, digamos, em um cΓclotron, verifica-se que este se comporta como se tivesse uma massa muito maior do que a sua massa de repouso. Quanto mais prΓ³ximo este encontrar-se da velocidade da luz, maior serΓ‘ sua inΓ©rcia, ou seja, sua massa, pois tambΓ©m maior Γ© a sua energia cinΓ©tica (aqui, necessariamente relativΓstica), e o que Γ© mais importante, maior serΓ‘ a quantidade de energia a ser acrescida para que este apresente uma mesma variação de velocidade. No limite em que este se move praticamente Γ velocidade da luz, sua massa Γ© infinitamente grande, e uma quantidade de energia infinita teria que ser-lhe acrescida para fazΓͺ-lo finalmente chegar Γ velocidade da luz.
Energia radiante
Trata-se de energia pura propagando-se pelo espaΓ§o em forma de ondas associadas a um campo. Γ, em vista do paradigma moderno, a energia diretamente associada Γ radiação eletromagnΓ©tica: Γ luz, Γ s ondas de rΓ‘dio, aos raios infravermelhos, aos raios X, e outras.
A
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] | 1,024 |
Wikipedia
| 0.1794 | 1.194606 | -0.433842 | 0.350118 |
energia radiante atravessa perfeitamente o vΓ‘cuo: a quase totalidade de energia que recebemos do sol chega atΓ© nΓ³s na forma de energia radiante distribuΓda em uma larga faixa de frequΓͺncias, faixa esta que inclui a faixa do visΓvel na regiΓ£o de maior densidade de energia, com as diversas cores (violeta, azul, verde, amarelo, laranja, vermelho) que conseguimos enxergar sendo particularmente intensas no espectro solar. Contudo o homem nΓ£o se restringiu a usar apenas os olhos para vasculhar o cosmo; radiotelescΓ³pios observam o cosmos em comprimentos de onda que nΓ£o podemos ver, indo desde as ondas de rΓ‘dio atΓ© os raios X e mesmo raios cΓ³smicos.
As ondas eletromagnΓ©ticas sΓ£o uma combinação de campos magnΓ©tico e elΓ©tricos ortogonais variΓ‘veis que sustentam-se mutuamente mediante da lei da indução de Faraday e a Lei de AmpΓ¨re em sua forma generalizada por Maxwell, possuindo, uma vez produzidas, existΓͺncias independentes das cargas aceleradas que a geraram. Ressalta-se que "cargas estΓ‘ticas e cargas em movimento com velocidade (vetorial) constante nΓ£o irradiam. Cargas aceleradas irradiam.".
Observe que, embora não irradiem ondas eletromagnéticas, cargas elétricas estÑticas e cargas em movimento não acelerado possuem seus campos elétricos e no último caso também magnéticos associados, e nestes campos hÑ energia armazenada. Contudo estes campos e estas energias estão "presos" à carga, e não propagando-se livremente pelo espaço, como ocorre com a energia nas ondas eletromagnéticas. Aos campos das cargas nestas condiçáes associam-se a energia potencial elétrica e a "energia magnética" antes referida no subtópico "Cargas elétricas em movimento" dentro do "Energia cinética" deste artigo.
A energia transportada em uma onda eletromagnética é removida da carga acelerada mediante um fenômeno conhecido por reação à radiação (fórmula de Larmor). Ondas eletromagnéticas não transportam apenas energia; transportam também momento. O fluxo de energia em uma onda eletromagnética é descrito pelo vetor de Poynting , cuja direção é perpendicular ao plano estabelecido pelos vetores campo elétrico e campo magnético , sendo obtido por:
onde representa a permeabilidade magnΓ©tica do vΓ‘cuo e "X" representa o produto vetorial.
Energia na fΓsica quΓ’ntica
A fΓsica quΓ’ntica prevΓͺ que a energia se manifesta em pequenos "pacotes" ou "quanta" de energia. Isto implica que a energia sΓ³ pode adotar valores discretos ("quantizados") e nΓ£o qualquer valor em uma escala contΓnua como era previsto pela mecΓ’nica clΓ‘ssica. Para os fenΓ΄menos macroscΓ³picos, essa caracterΓstica da energia nΓ£o apresenta diferenΓ§as significativas no comportamento dos sistemas, jΓ‘ que as quantidades de energia envolvidas sΓ£o grandes (compostas por uma quantidade enorme de "quanta") de modo que ela seja praticamente contΓnua. No caso do eletromagnetismo, os fΓ³tons sΓ£o os "quanta" de luz e a energia que carregam Γ© diretamente proporcional Γ frequΓͺncia de oscilação deles de acordo com a Relação de Planck-Einstein: E = hv, onde h Γ© a constante de Planck (hl) e v Γ© a frequΓͺncia.
Na mecΓ’nica quΓ’ntica, teoria que descreve formalmente e matematicamente o funcionamento de partΓculas ou sistemas nos domΓnios de validade da fΓsica quΓ’ntica, a energia total Γ© definida em termos do hamiltoniano, o operador de energia, e sua evolução no tempo Γ© dada pela equação de SchrΓΆdinger. Apesar
|
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Wikipedia
| 0.091223 | 0.898711 | -0.522159 | 0.205922 |
dessas diferenΓ§as, na fΓsica quΓ’ntica a energia possui as mesmas unidades de medida e obedece Γ s mesmas leis de conservação e de transformação postuladas pela fΓsica clΓ‘ssica.
Recursos energΓ©ticos
Energia solar
O termo energia solar refere-se Γ toda energia que tem origem no sol, sendo em quase sua totalidade representada pela energia radiante emitida por este corpo celeste. Uma pequena parcela desta energia encontra-se associada Γ energia cinΓ©tica transportadas pelo vento solar.
O sol Γ© a fonte primΓ‘ria de toda a energia que usamos na Terra excetuando-se a energia nuclear - com origem nos nΓΊcleos atΓ΄micos dos elementos quΓmicos, formado em estrelas antecedentes ao sol e que no processo de sua morte, liberaram ao espaΓ§o sideral o material que hoje encontramos aqui na Terra - e talvez parte da energia geotΓ©rmica - a parcela com origem na energia potencial gravitacional liberada no processo de agregação de matΓ©ria que formou o planeta e que, convertida em energia tΓ©rmica, incandesceu a Terra durante sua infΓ’ncia. Ademais, da energia hidrelΓ©trica Γ energia tΓ©rmica liberada pela combustΓ£o de combustΓveis fΓ³sseis e mesmo Γ energia quΓmica presente em uma pilha, todas remontam Γ energia solar em algum momento. Γ o sol que provΓͺ a energia necessΓ‘ria Γ evaporação da Γ‘gua, que, levada atravΓ©s de nuvens Γ s elevadas altitudes, precipita-se na cabeceira dos rios. Γ o sol que provΓͺ a energia necessΓ‘ria Γ fotossΓntese, sendo a fonte primΓ‘ria de toda a energia quΓmica armazenada nos seres vivos em virtude da cadeia alimentar, e nos combustΓveis fΓ³sseis, destes derivados.
O termo energia solar, em escopo moderno, pode referir-se ao processo de captação de energia via placas solares, onde a energia radiante é diretamente convertida em energia elétrica, e também ao processo de aquecimento de Ñgua via coletores solares, o que evita gastos com a compra de energia elétrica a fim de aquecer-se Ñgua para o uso humano.
Energia elΓ©trica
A chamada energia elΓ©trica nada mais Γ© do que a energia potencial elΓ©trica associada a um sistema onde uma determinada carga elΓ©trica encontra-se situada nΓ£o em um condutor elΓ©trico de referΓͺncia - onde define-se a energia potencial desta como sendo nula - mas em um segundo condutor de eletricidade que geralmente acompanha o primeiro mas encontra-se deste isolado. Esta carga, ao passar do fio onde se encontra para o fio de referΓͺncia libera a energia potencial a ela associada, sendo esta convertida em energia tΓ©rmica (em um chuveiro, via efeito joule), energia radiante (em um forno microondas), energia cinΓ©tica (em um motor), ou outra forma de energia qualquer no interior do componente que permitiu sua passagem de um fio a outro. Explica-se assim porque as tomadas de energia tΓͺm sempre no mΓnimo dois fios.
AnΓ‘lise detalhada deste sistema leva-nos diretamente ao conceito de energia potencial elΓ©trica jΓ‘ previamente considerado neste artigo e a uma Γ‘rea de estudos especΓfica dentro da fΓsica: a anΓ‘lise de circuitos, esta sempre presente mesmo nos piores cursos de eletrΓ΄nica. O leitor Γ© remetido aos tΓ³picos especΓficos para maiores detalhes.
Energia hidrelΓ©trica
A energia hidrelΓ©trica Γ© a energia que vem do movimento das Γ‘guas, usando o potencial hidrΓ‘ulico de um rio de nΓveis naturais,queda d'Γ‘gua naturais ou artificiais. Essa energia Γ© a segunda maior fonte de eletricidade do mundo. Frequentemente constroem
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Wikipedia
| -0.128986 | 0.327685 | -0.618544 | 0.398184 |
-se represas que reprimem o curso da Γ‘gua, fazendo com que ela se acumule em um reservatΓ³rio denominado barragem. Toda a energia elΓ©trica gerada dessa maneira Γ© levada por cabos, dos terminais do gerador elΓ©trico atΓ© os transformadores elΓ©tricos e entΓ£o ao usuΓ‘rio final. A energia hidrelΓ©trica apresenta certos problemas, como consequΓͺncias socioambientais de alagamentos de grandes Γ‘reas, alteração do clima, fauna e flora locais, dentre outros. Entretanto ainda Γ©, se comparado a outras, uma forma limpa de se gerar energia para o consumo humano.
Energia hidrelΓ©trica no Brasil: devido Γ sua enorme quantidade de rios, a maior parte da energia elΓ©trica disponΓvel Γ© proveniente de grandes usinas hidrelΓ©tricas. A energia primΓ‘ria de uma hidrelΓ©trica Γ© a energia potencial gravitacional da Γ‘gua contida numa represa elevada. Antes de se tornar energia elΓ©trica, a energia primΓ‘ria deve ser convertida em energia cinΓ©tica de translação da Γ‘gua e posteriormente em energia cinΓ©tica de rotação no gerador elΓ©trico. O dispositivo que realiza esta ΓΊltima transformação Γ© a turbina. Ela consiste basicamente em uma roda dotada de pΓ‘s, que Γ© posta em rΓ‘pida rotação ao receber o impulso da massa de Γ‘gua. O ΓΊltimo elemento dessa cadeia de transformaçáes Γ© o gerador, que converte o movimento rotatΓ³rio da turbina em energia potencial elΓ©trica.
Energia quΓmica
Γ o nome da energia que estΓ‘ armazenada nas ligaçáes covalentes, iΓ΄nicas, metΓ‘licas, ou de forma similar em qualquer das ligaçáes responsΓ‘veis pela estrutura da matΓ©ria conforme a concebemos hoje. Em essΓͺncia Γ© a energia potencial elΓ©trica associada Γ s posiçáes relativas dos elΓ©trons nos orbitais eletrΓ΄nicos (dos elΓ©trons - negativos) e dos nΓΊcleos atΓ΄micos (positivos) uns em relação aos outros, recebendo este nome em particular apenas para enfatizar a ordem de grandeza e as partΓculas constituintes do sistema em estudo, composto por Γ‘tomos, molΓ©culas e/ou Γons em interação, que pode ser liberada ou armazenada mediante reaçáes quΓmicas.
Em uma reação quΓmica os nΓΊcleos alteram suas posiçáes uns em relaçáes aos outros, bem como os orbitais eletrΓ΄nicos presentes nas eletrosferas atΓ΄micas tambΓ©m o fazem, sobretudo os orbitais associados Γ ΓΊltima camada eletrΓ΄nica de cada Γ‘tomo, na conhecida camada de valΓͺncia. Este rearranjo pode levar a uma configuração espacial final com uma energia potencial maior do que na configuração inicial -no caso das reaçáes endoenergΓ©ticas - ou a uma configuração espacial com menor energia potencial elΓ©trica em relação Γ inicial, caso em que a diferenΓ§a Γ© geralmente convertida em energia tΓ©rmica - o que aumenta a temperatura do sistema - e posteriormente liberada Γ s vizinhanΓ§as do sistema, devido ao aumento de temperatura, na forma de calor.
Assim existem ligaçáes as quais se associa grande quantidade de energia quΓmica e ligaçáes as quais se associa uma quantidade bem menor de energia quΓmica. A Γ‘gua Γ© um exemplo de molΓ©cula com ligaçáes H-O, pobres em energia quΓmica se comparadas Γ s ligaçáes H-H e O=O. A reação entre H2 e O2 leva a uma reestruturação espacial na qual parte da energia quΓmica dos reagentes Γ© liberada: a formação de vapor de Γ‘gua a partir dos gases reagentes Γ© em verdade uma reação exoenergΓ©tica explosiva.
Em biologia
De importÒncia dentro da biologia destaca-se não só a Ñgua como a glicose, rica em ligaçáes H-C e outras, que se comparadas à ligaçáes C=O presente no CO2 e H-O presente na Ñgua
|
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Wikipedia
| -0.170761 | 0.709234 | -0.352475 | 0.266636 |
, possuem maior energia quΓmica associada. Ao passo que a sΓntese da glicose a partir do CO2 e H2O Γ© portanto uma reação endoenergΓ©tica, sendo realizada no processo de fotossΓntese nas plantas Γ s expensas da energia radiante recebida do sol, a combustΓ£o da glicose, representada pelo processo inverso, constitui a principal fonte de energia dos seres vivos aerΓ³bicos.
Os seres vivos aerΓ³bicos utilizam a glicose como principal combustΓvel (fonte de energia quΓmica); entretanto, esta molΓ©cula nΓ£o pode ser utilizada diretamente, pois sua quebra direta libera de forma imediata muito mais energia que o necessΓ‘rio para o trabalho celular. Uma tora de madeira a arder em chamas Γ© uma amostra desta capacidade de conversΓ£o de energia. Por isso a natureza selecionou mecanismos mais controlados, que incluem a transferΓͺncia da energia quΓmica da glicose para molΓ©culas tipo ATP (adenosina trifosfato) antes de seu uso final. Nos primeiros seres vivos a habitarem o planeta surgiu o primeiro destes mecanismos com tal objetivo: a fermentação. A fermentação anaerΓ³bia, alΓ©m do ATP, gera tambΓ©m etanol e diΓ³xido de carbono (CO2). A presenΓ§a de CO2 na atmosfera possibilitou o surgimento da fotossΓntese. Este processo fez surgir o O2 (oxigΓͺnio) na atmosfera. Com o oxigΓͺnio, outros seres vivos puderam desenvolver um novo mecanismo de transferΓͺncia de energia quΓmica da glicose para o ATP: a respiração aerΓ³bica. Ao longa da histΓ³ria do planeta a mudanΓ§a na atmosfera, ao tornar-se rica em O2, foi responsΓ‘vel por propiciar uma explosΓ£o na diversidade de seres a utilizarem a respiração aerΓ³bica como mecanismo de obtenção de energia; este perΓodo da evolução ficou conhecido nos anais da biologia por explosΓ£o cambriana.
Nos organismo biolΓ³gicos a energia quΓmica pode ser diretamente transformada em energia cinΓ©tica (nos mΓΊsculos) ou tΓ©rmica, sendo esta de grande importΓ’ncia para os organismos homeotΓ©rmicos.
EletroquΓmica
A energia quΓmica pode ser transformada diretamente em outras formas de energia que nΓ£o Γ© tΓ©rmica, por exemplo em eletricidade (nas baterias ou nas cΓ©lulas de hidrogΓͺnio em automΓ³veis modernos). HΓ‘ uma Γ‘rea da quΓmica especialmente destinada a este estudo, a eletroquΓmica.
CombustΓveis
O petrΓ³leo e demais combustΓveis fΓ³sseis como o carvΓ£o mineral tΓͺm relevΓ’ncia inegΓ‘vel na modernidade. Representam uma considerΓ‘vel parcela da matriz energΓ©tica em nossa sociedade atual, e constituem motivo de preocupação, entre outros, por nΓ£o serem renovΓ‘veis. HΓ‘ ainda o problema do aquecimento global, diretamente relacionado aos mesmos. A busca de combustΓveis alternativos, como o etanol, evidencia a importΓ’ncia dos combustΓveis em nossa sociedade, assim como a importΓ’ncia dos problemas associados Γ sua produção, distribuição e consumo.
Energia eΓ³lica
A energia eΓ³lica tem sido aproveitada desde a antiguidade para mover os barcos impulsionados por velas ou para fazer funcionar a engrenagem de moinhos, ao mover as suas pΓ‘s. Nos moinhos de vento a energia eΓ³lica era transformada em energia mecΓ’nica, utilizada na moagem de grΓ£os ou para bombear Γ‘gua. Os moinhos foram usados para fabricação de farinhas e ainda para drenagem de canais, sobretudo nos PaΓses Baixos.
Na atualidade utiliza-se a energia eΓ³lica para mover aerogeradores - grandes turbinas colocadas em lugares de muito vento. Essas turbin
|
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Wikipedia
| -0.016568 | 0.597624 | -0.17248 | 0.499374 |
as tΓͺm a forma de um catavento ou um moinho. Esse movimento, atravΓ©s de um gerador, produz energia elΓ©trica.
A energia eólica vem gradualmente ganhando importÒncia em vista das preocupaçáes modernas no que refere-se a fontes de energias limpas e renovÑveis.
O auto Γ© remetido ao artigo principal para maiores detalhes.
Energia nuclear
Conforme visto, a energia potencial nuclear é a energia potencial associada à posição relativa dos nucleôns uns em relação aos outros em virtude da interação nuclear forte que os mantém unidos no núcleo.
A variação da energia potencial nuclear durante o processos de reação nuclear em um Γ‘tomo Γ© geralmente enorme se comparada Γ s variaçáes de energia quΓmica encontradas quando este mesmo Γ‘tomo participa de reaçáes quΓmicas as mais exoenergΓ©ticas (da ordem de centenas a milhares de vezes maior). Os processos nucleares que liberam energia sΓ£o assim extremamente exoenergΓ©ticos, e pequenas quantidades de material reativo podem liberar quantidades astronΓ΄micas de energia.
As reaçáes nucleares exoenergΓ©ticas sΓ£o geralmente a fissΓ£o de Γ‘tomos com grandes nΓΊcleos (onde destacam-se como elemento natural os isΓ³topos do urΓ’nio e como elemento jΓ‘ artificial os isΓ³topos do plutΓ΄nio) ou a fusΓ£o de Γ‘tomos com nΓΊcleos pouco massivos (com destaque para os isΓ³topos do hidrogΓͺnio). AtΓ© os dias de hoje, embora haja considerΓ‘vel pesquisa associada ao processo de fusΓ£o, apenas a energia liberada atravΓ©s dos processos de fissΓ£o Γ© praticamente utilizΓ‘vel. A energia que liberam Γ© transformada sobretudo em energia cinΓ©tica presente nas radiaçáes alfa ou beta, em energia radiante associada Γ radiaçáes gama e em energia tΓ©rmica que eleva de forma considerΓ‘vel a temperatura da amostra em reação, podendo facilmente vir a fundi-la em processos ainda longe do crΓtico (explosivo). Sob controle em um reator nuclear, esta energia tΓ©rmica liberada pode ser convertida em energia elΓ©trica mediante emprego da mesma tecnologia usada nas termoelΓ©tricas: muda-se apenas a fonte de energia primΓ‘ria, que passa a ser o reator nuclear ao invΓ©s da fornalha quΓmica). Sem controle, uma pequena quantidade de material reativo podem gerar uma explosΓ£o monumental, o que, levado a cabo, deu origem Γ s ditas armas nucleares.
Em termos histΓ³ricos, o domΓnio do processo de fusΓ£o Γ© posterior ao domΓnio do processo de fissΓ£o atΓ΄mica, pois precisa-se da energia liberada na fissΓ£o para iniciar-se o processo de fusΓ£o, pelo menos aqui na Terra. A energia que recebemos do sol tem sua origem no processo de fusΓ£o nuclear de Γ‘tomos de hidrogΓͺnio, constantemente convertidos em hΓ©lio no nΓΊcleo desta estrela, sendo este processo tambΓ©m extremamente exoenergΓ©tico se comparado a uma reação quΓmica convencional. Isto justifica em parte o maior poder destrutivo de uma bomba nuclear de hidrogΓͺnio (bomba H) tendo em vista a facilidade de obtenção de seus isΓ³topos deutΓ©rio e trΓtio se comparada Γ dificuldade de obtenção de U235 a exemplo. O processo de fissΓ£o nuclear do urΓ’nio, descoberta em 1939 pelos cientistas alemΓ£es Otto Hahn, Lise Meitner e Fritz Strassmann ao bombardearem Γ‘tomos de urΓ’nio com nΓͺutrons mediante a observação de que estes entΓ£o se dividiam em dois fragmentos, na maioria dos casos em estrΓ΄ncio e xenΓ΄nio ou em criptΓ΄nio e bΓ‘rio, com liberação de mais dois ou trΓͺs nΓͺutrons energΓ©ticos, nΓ£o teria saΓdo dos limites estritos do laboratΓ³rio se nΓ£o fosse pelo fato de que neste processo hΓ‘ tambΓ©m a liber
|
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Wikipedia
| -0.31918 | 0.652131 | -0.177411 | -0.171011 |
*Vidal-Hall and others v Google Inc (The Information Commissioner intervening)
Data protection Processing of information. Google had sought, to set aside the permission that had been granted to the claimants to serve their claim form out of the jurisdiction in their action which alleged misuse of private information, breach of confidence and breach of the (the DPA). The action for breach of confidence was set aside. The Court of Appeal, Civil Division, dismissed Google's appeal as the pleaded actions were clearly arguable and not pointless. The court held that misuse of private information should be recognised as a tort for the purposes of service out of the jurisdiction and that, in order to make s13(2) of the DPA compatible with EU law, that section had to be disapplied, with the consequence that compensation would be recoverable under s13(1) for any damage suffered as a result of a contravention by a data controller of the requirements of the DPA.
*R (on the application of Evans) and another v Attorney General (Campaign for Freedom of Information Intervening)
Freedom of information Exempt information. The Attorney General appealed against the decision of the Court of Appeal, Civil Division, quashing a certificate issued pursuant to s53(2) of the and reg18(6) of the Environmental Information Regulations 2004, SI2004-3391, in respect of the disclosure of communications passing between The Prince of Wales and ministers in various government departments. The Supreme Court, in dismissing the appeal, held that the communications requested were not excepted from any duty of disclosure to the claimant under s53 of the Act. Further, the effect of Directive (EC) 2003-4 (on public access to environmental information and repealing Council Directive 90-313-EEC) was to invalidate the certificate in relation to the environmental information, but not in relation to the non-environmental information in the advocacy correspondence.
RTA (Business Consultants) Ltd v Bracewell
Contract Illegality. The claimant carried on business as business transfer agents. It was common ground that the activities of the claimant fell within the definition of the expression 'estate agency work' in of the Estate Agents Act 1979. The relevance of that was the reference to 'estate agents' in the Money Laundering Regulations 2007, . The parties entered into an agreement and a dispute arose. The claimant brought a claim based on the agreement. The defendant contended that the agreement was not enforcible for illegality. Applying established law, the Queen's Bench Division held that the agreement was illegal and was consequently unenforceable.
*R (on the application of Catt) v Metropolitan Police Commissioner; R (on the application of T) v Metropolitan Police Commissioner
Human rights Right to respect for private and family life. The present appeals concerned the systematic collection and retention by police authorities of electronic data about individuals. The Supreme Court, in allowing the Metropolitan Police Commissioner's appeals, held that there had been no disproportionate interference with the respondents' rights under art8 of the European Convention on Human Rights. Retention of material concerning the first respondent was justified by the legitimate requirements of police intelligence gathering and, as to the second respondent, the retention policy had been flexible enough to allow for information to be deleted when retaining it would no longer serve any useful policing purpose.
Crook v Chief Constable of Essex Police
Equity Breach of confidence. The claimant brought a claim for damages against the defendant Chief Constable of Essex Police arising from a press release by Essex Police. The Queens Bench Division held that the release of the information had not been reasonably necessary and proportionate in all the circumstances of the case and the claimant was awarded damages, including 57,750 special damages for loss of earnings.
*Re Law Society (Solicitors Regulation Authority)
Solicitor Law Society. The Chancery Division held that the Law Society had the power to destroy old and redundant documents seized in connection with interventions in solicitors' practices, such power being justified as falling within the scope of para 16 of Pt II of to the Solicitors Act 1974.
Deer v University of Oxford
Employment tribunal Striking out. The employee had previously settled a claim against the employer. She then brought proceedings alleging victimisation
|
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CommonCrawl
| -0.373861 | 1.03368 | 0.257445 | 0.246398 |
Global Fund has awarded $1 billion to support countries' COVID-19 responses, but funding for this purpose is now fully deployed
Abbie Minter 2020-12-23T15:49:08+11:00 December 23rd, 2020|
GENEVA β The Global Fund has awarded nearly US$1 billion to 106 countries to support their responses to COVID-19, but has now fully deployed all its funding for this purpose. There are significant further needs for immediate funding, including for personal protective equipment (PPE), testing and treatment, and to mitigate the impact on lifesaving HIV, TB and malaria programs. Unfunded country requests for support now amount to over US$355 million.
In response to COVID-19 the Global Fund has awarded an additional US$980 million to 106 low- and middle-income countries and 14 multi-country programs in 2020. The final US$41.5 million was approved yesterday evening. The awards for COVID-19 are on top of the approximately US$4 billion the Global Fund has invested in its core HIV, TB and malaria programs this year.
Countries are using the funds awarded as part of the Global Fund's COVID-19 response to:
Reinforce national COVID-19 responses, including purchasing critical tests, treatments and medical supplies; protecting front-line health workers with training and PPE like gloves and masks; and supporting control and containment interventions, including test, trace and treat/isolate;
Mitigate COVID-19 impact on lifesaving HIV, TB and malaria programs, including by delivering medicines, mosquito nets and critical supplies door to door, protecting community health workers and providing support and prevention services via digital platforms;
Make urgent improvements to health and community systems to help fight COVID-19, HIV, TB and malaria, including by reinforcing supply chains, laboratory networks and community-led response systems.
The Global Fund partnership moved swiftly to help countries respond to COVID-19, providing millions of tests and PPE, and enabling rapid adaption of HIV, TB and malaria programs. In addition to redeployed internal funds, the Global Fund's COVID-19 response received US$259 million from donors including Canada, Denmark, Germany, Italy, Sweden, Norway and the FIFA Foundation.
"We are extremely grateful to the donors who, even while fighting the virus at home, contributed extra funding to fight COVID-19 in the world's poorest and most vulnerable countries," said Peter Sands, Executive Director of the Global Fund. "We are struck by what countries and communities have accomplished to fight the new virus and protect hard-won gains in the fight against HIV, TB and malaria. But the tragedy is that just as new COVID-19 tools are becoming available, and as needs are on the rise, there is no money left on the table. Cefixime is an antibiotic in the form of pills and suspensions that destroys streptococci, protea, moscarella, salmonella, klebsiella. It is prescribed for children (from six months) and adults with bronchitis, otitis, pharyngitis, sinusitis, tonsillitis as well as for pathologies of the urinary tract. For more information about the drug, go to https://mypts.com/online-antibiotics/.
"While the fantastic news on vaccines provides a light at the end of the tunnel, we must not let this blind us to the reality that the tunnel ahead remains long, dark and dangerous, particularly for the poorest and most vulnerable communities," Sands continued. "We must continue to step up investment in testing, treatment, and PPE. And we must recognize that in some countries, the knock-on impact of the pandemic on HIV, TB and malaria may exceed the direct impact."
The Global Fund has estimated that it needs a further US$5 billion on top of its core funding to support countries in responding to the pandemic. This figure represents part of the overall financing needs of the Access to COVID-19 Tools Accelerator (ACT-Accelerator), the global collaborative partnership in which the Global fund plays a leading role.
The Global Fund is extremely appreciative of the continued support of donors for its core funding, as pledges made at the record-breaking Re
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CommonCrawl
| 0.731643 | 0.278369 | 1.324413 | 0.886426 |
Archeologist Khaled Asaad Refused To Tell Location...
Say nay
Respected Contributor
Archeologist Khaled Asaad Refused To Tell Location Of Hidden Antiquities
GRAPHIC. DON'T READ IF YOU'RE SENSITIVE.
This is absolutely heinous. May he rest in peace.
From Huffpo:
The 82-year-old scholar worked for over five decades as the head of antiquities at the ancient site.
He was nicknamed "Mr. Palmyra" after the ancient Syrian city whose excavation and preservation he had dedicated most of his life to.
Family, friends and colleagues say it was this very dedication that cost 82-year-old Khaled Asaad his life. The renowned Syrian archeologist was beheaded by Islamic State (ISIS) militants this week. His bloodied body was left to hang on one of the Roman columns in a main square of the historic site.
According to reports, Asaad was murdered Tuesday in front of dozens of witnesses after refusing to reveal to ISIS militants the location of valuable Palmyran artifacts that had been moved for safekeeping.
Chris Doyle, director of the Council for Arab-British Understanding, told The Guardian that the archeologist, who had been held by ISIS for more than a month, had been interrogated about the ancient treasures and had been "executed when he refused to cooperate."
Hundreds of artifacts and statues from the Palmyra museum had reportedly been moved to a safe place before ISIS seized the ruins last spring. Asaad, who had been in charge of Palmyra's archeological site for 40 years before he retired in 2003, played a central role in the move, the Associated Press reports.
The ruins, a UNESCO heritage site, dates back some 2,000 years.
ISIS insurgents, who have imposed a puritanical and violent interpretation of Islamic law across the swathes of Syria and Iraq that it now controls, have been known to destroy ancient relics, which they claim promote idolatry. It's believed that the extremist group also loots and sells antiquities to fund its activities.
Footage emerged earlier this year of ISIS destroying statues in Mosul. ISIS soldiers have reportedly also destroyed buildings and artifacts in the ancient Assyrian cities of Ninevah and Nimrud in Iraq.
Asaad's connection with Palmyra meant that he was a prime target for the jihadi group. His family and friends say, however, that he refused to leave his home and the ancient city he loved so dearly.
"I begged him two months ago to leave the town and come to Damascus with his family, but he refused," Ahmad Ferzat Taraqji, an antiquities expert and friend of Asaad's, told the AP. "He believed in destiny. He told me, 'I was born in Palmyra and will stay in Palmyra and will not leave even if costs me my blood.'"
This week, the archeological community has been reeling in the wake of Asaad's death, whose depth of knowledge of Palmyra -- which in antiquity had been an important trading post along the Silk Road -- has been called "irreplaceable."
Asaad, who studied history and education at the University of Damascus, penned many scholarly works on Palmyra in his long career. Syrian state news agency Sana said he also discovered several ancient graveyards, caves and a Byzantine cemetery in the garden of Museum of Palmyra.
Asaad was also a scholar of the ancient language Aramaic.
"He was Mr. Palmyra, you couldn't do any work in Palmyra without going through him," Amr al-Azm, an antiquities expert and professor at Shawnee State University in Ohio, told the AP. "No one's been there for consistently so long and covered so many aspects of Palmyra's cultural heritage. I think it's irreplaceable."
State Department spokesman John Kirby said this week that the U.S. "condemns in the strongest possible terms this murder ... of a man who dedicated his life to preserving Syria's cultural treasures."
"[Asaad's] life
|
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CommonCrawl
| 0.87293 | 0.818248 | 2.253303 | 0.711873 |
\section{Introduction}
The search for free objects in division rings has been largely motivated by the
following two conjectures that still remain open:
\begin{itemize}
\item[(G)] If $D$ is a noncommutative division ring, then the
multiplicative group $D\setminus\{0\}$ contains a free group of rank two;
\item[(A)] If $D$ is a division ring which is infinite dimensional over its center $Z$
and it is finitely generated
(as a division algebra over $Z$), then $D$ contains a free $Z$-algebra of rank two.
\end{itemize}
Conjecture (G) was stated by A. I. Lichtman in \cite{Lichtmanfreesubgroupsof} and it has been proved
when the center of $D$ is uncountable \cite{chibafreegroupsinsidedivisionrings} and
when $D$ is finite dimensional over its center \cite{Goncalvesfreegroupsinsubnormal} to name
two important instances where it holds true.
Conjecture (A) was formulated independently
by L. Makar-Limanov in \cite{Makaronfreesubobjects} and T. Stafford. Evidence for conjecture (A)
has been provided in many papers, for example \cite{Makar1}, \cite{Makar2}, \cite{Lichtmanfreeuniversalenveloping},
\cite{BellRogalskifreesubalgebrasoreextensions}, \cite{BellGoncalvesOreexentions}.
In many division rings in which conjecture (A) holds, $D$ in fact contains a
noncommutative free group $Z$-algebra. For example, this always happens
if the center of $D$ is uncountable~\cite{GoncalvesShirvani} (or \cite{SanchezObtaininggraded}
for a slightly more general result).
Other examples of the
existence of free group algebras in division rings
can be found in \cite{Cauchoncorps}, \cite{Makar-LimanovOnsubalgebrasofthe},
\cite{Lichtmanfreeuniversalenveloping},
\cite{SanchezfreegroupalgebraMNseries}.
Therefore it makes sense to consider the following unifying
conjecture:
\begin{itemize}
\item[(GA)] Let $D$ be a skew field with center $Z$.
If $D$ is finitely generated as a division ring over $Z$ and $D$ is
infinite dimensional over $Z$, then $D$ contains a noncommutative
free group $Z$-algebra.
\end{itemize}
For more details on these and related conjectures the reader is
referred to \cite{GoncalvesShirvaniSurvey}.
\medskip
After the work in \cite{GoncalvesShirvaniSurvey},
\cite{FerreiraGoncalvesHeisenberggroup}, \cite{FerreiraGoncalvesSanchezFreegroupssymmetric},
\cite{FerreiraFornaroliGoncalvesFreeinvolution},
\cite{FerreiraGoncalvesSanchez1}, \cite{FerreiraGoncalvesSanchez2} it has become apparent that
an involutional version of conjectures (G) and (A) should be investigated. Part of our work
deals with an involutional version of (GA).
To be more specific, if
$D$ is equipped with an involution,
under the hypothesis of (GA), can we find a free group algebra whose free
set of generators is formed by symmetric elements (i.e. $x^*=x$)?
\medskip
Let $k$ be a field. A \emph{$k$-involution} on a $k$-algebra $R$ is a
$k$-linear map $*\colon R\rightarrow R$, $x\mapsto x^*$, such that
$(ab)^*=b^*a^*$ for all $a,b\in R$. There are two families of $k$-algebras that usually are equipped with an involution.
These are group $k$-algebras and universal enveloping algebras of Lie $k$-algebras.
Given an involution on a group (see p.\pageref{involutiongroup}
|
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] | 1,024 |
ArXiv
| 0.538732 | 2.362612 | 0.586502 | 1.159639 |
for precise a definition),
it induces a $k$-involution on the
group $k$-algebra $k[G]$ (p.\pageref{involutiongroup}). Furthermore, if $G$ is an orderable group
(p.\pageref{orderablegroup}), there is a prescribed
construction of a division $k$-algebra, which we call $k(G)$, that contains $k[G]$, it
is generated by $k[G]$ and such that any $k$-involution on $k[G]$ can be
extended to $k(G)$ (see Section~\ref{sec:Heisenberggroup} for more details).
Also, given a $k$-involution (see p.\pageref{kinvolutionLie}) of a Lie $k$-algebra $L$,
it induces a $k$-involution on the universal enveloping algebra $U(L)$ in the natural way
(\cite[Section~2.2.17]{Dixmierenvelopingalgebras}). There is also a concrete construction
of a division $k$-algebra, which we denote by $\mathfrak{D}(L)$. It contains
$U(L)$, it is generated by $U(L)$ and such that any
$k$-involution on $L$ can be extended to a $k$-involution of $\mathfrak{D}(L)$ (see Section~\ref{sec:residuallynilpotent}
for more details).
We remark that neither $k[G]$ nor $U(L)$ need to be Ore domains, but
if they are, both $k(G)$ and $\mathfrak{D}(L)$ coincide with the Ore ring of fractions
of $k[G]$ and $U(L)$ respectively.
\medskip
The aim of our work is to apply the graded and filtered methods developed in \cite{SanchezObtaininggraded}
and \cite{SanchezObtainingI} to obtain free group algebras in division rings.
Concerning conjecture (GA), we are able to prove an extension of
a result by Lichtman.
More precisely, \cite[Theorem~4]{Lichtmanfreeuniversalenveloping} is
(2) of the following result.
\begin{theo}\label{theo:Lichtman}
Let $k$ be a field of characteristic zero and $L$ be a nonabelian Lie $k$-algebra.
If one of the following conditions is satisfied
\begin{enumerate}[\rm(1)]
\item $L$ is residually nilpotent;
\item The universal enveloping algebra $U(L)$
is an Ore domain;
\end{enumerate}
then $\mathfrak{D}(L)$ contains a (noncommutative) free group
$k$-algebra. \qed
\end{theo}
Notice that $\mathfrak{D}(L)$ may not contain a free $k$-algebra of rank two if the characteristic of $k$
is not zero. In fact, as noted in \cite[p.147]{Lichtmanfreeuniversalenveloping}, the
proof given in \cite[p.204]{JacobsonLiealgebras} shows that if $L$ is finite dimensional over $k$,
then $\mathfrak{D}(L)$ is finite dimensional over its center. Therefore, it does not contain a
noncommutative free algebra.
\medskip
Concerning involutional versions of conjecture (GA), we are able to prove the following two theorems.
\begin{theo}\label{theo:intro2}
Let $k$ be a field of characteristic zero and $L$ be a nonabelian Lie $k$-algebra
endowed with a $k$-involution $*\colon L\rightarrow L$. Suppose that one of the following
conditions is satisfied.
\begin{enumerate}[\rm(1)]
\item $L$ is residually nilpotent;
\item The universal enveloping algebra $U(L)$ is an Ore domain and either
\begin{enumerate}[\rm(a)]
\item there exists $x\in L$ such that $[x^*,x]\neq 0$ and the Lie $k$-subalgebra
of $L$ generated by $\{x,x^*\}$ is of dimension at least three, or
\item $[x^*,x]=0$ for every $x\in L$, but
|
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] | 1,024 |
ArXiv
| 0.400841 | 2.487199 | -0.185481 | 0.84089 |
there exist $x,y\in L$ with $[y,x]\neq 0$
and the $k$-subspace of $L$ spanned by $\{x,x^*,y,y^*\}$ is not equal to the
Lie $k$-subalgebra of $L$ generated by $\{x,x^*,y,y^*\}$.
\end{enumerate}
\end{enumerate}
Then $\mathfrak{D}(L)$ contains a (noncommutative) free group $k$-algebra whose free
generators are symmetric with respect to the extension of $*$ to $\mathfrak{D}(L)$. \qed
\end{theo}
\begin{theo}\label{theo:residuallynilpotentgroup}
Let $k$ be a field of characteristic zero and $G$ be a nonabelian residually torsion-free nilpotent
group endowed with an involution $*\colon G\rightarrow G$. Then $k(G)$
contains a free group $k$-algebra whose free generators are symmetric with respect to the
extension of $*$ to $k(G)$. \qed
\end{theo}
Notice that, since the map $L\mapsto L$, $x\mapsto -x$, is a $k$-involution
for any Lie $k$-algebra $L$, then Theorem~\ref{theo:intro2}(1) implies Theorem~\ref{theo:Lichtman}(1).
On the other hand, the proofs and the
elements that generate the free group algebra in Theoren~\ref{theo:intro2}
are more complicated than those of Theorem~\ref{theo:Lichtman}.
Let $k$ be a field of characteristic zero.
The general strategy to obtain Theorems~\ref{theo:Lichtman}
and \ref{theo:intro2} goes back to Lichtman~\cite{Lichtmanfreeuniversalenveloping},
and it was also used in \cite{FerreiraGoncalvesSanchez2}.
Roughly speaking, one has to obtain free (group) algebras in the division ring
$\mathfrak{D}(H)$, where $H$ is the
Lie $k$-algebra $H=\langle x,y\colon
[y,[y,x]]=[x,[y,x]]=0\rangle$. From this, one obtains free group algebras in
$\mathfrak{D}(L)$ where $L$ is a residually nilpotent Lie $k$-algebra. Now
there is a way to obtain free (group) algebras
in $\mathfrak{D}(L)$, where $L$ is a Lie $k$-algebra
such that $U(L)$ is an Ore domain, from the residually nilpotent case using filtered and graded methods.
We have improved and somehow clarified this strategy in order to obtain the two first theorems above.
Then
Theorem~\ref{theo:residuallynilpotentgroup} is obtained from the previous results using the filtered
methods from \cite{SanchezObtainingI} and a technique from \cite{FerreiraGoncalvesSanchez2}.
\bigskip
We begin Section~\ref{sec:generalfiltrations} introducing some basics on filtrations and valuations. In
Section~\ref{sec:filtrationuniversal}, we state some results on how filtrations and gradations of Lie algebras
induce filtrations and gradations of their universal enveloping algebras. Section~\ref{sec:freegroupalgebrasdivision}
is devoted to results about the existence of free group algebras obtained in \cite{SanchezObtaininggraded} and \cite{SanchezObtainingI}. They
show different ways of obtaining free group algebras in division rings generated by group graded rings,
and in division rings endowed with a valuation.
The results in Section~\ref{sec:filtrations&valuations} are stated in more generality than necessary in
subsequent sections, but we believe there is some merit in the general statements and they could be of interest to others.
The first part of Section~\ref{sec:nilpotentLieinvolutions} is concerned with the classifications of all the
$k$-involutions of the Heisenberg Lie $k$-algebra $H=\langle x,y\colon [y,[y,x]]=[x,[y,x]]
|
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] | 1,024 |
ArXiv
| 0.2171 | 2.248407 | -0.517675 | 0.632186 |
=0 \rangle$
over $k$, a field of characteristic different from two. We are able to prove that, up to equivalence,
there are three involutions on $H$. We then use this to show that any nilpotent Lie $k$-algebra
endowed with an involution $*\colon L\rightarrow L$ contains a $*$-invariant $k$-subalgebra
$H$ of $L$ and whose restriction to $H$ is one of those three involutions.
Section~\ref{sec:nilpotentOre} deals with the problem of finding free (group) algebras
in the Ore ring of fractions of $U(L)$, the universal enveloping algebra of a nilpotent
Lie $k$-algebra $L$ over a field of characteristic zero. The main result is the technical
Theorem~\ref{theo:freegroupHeisenberg}, where a lot of free (group) algebras in $\mathfrak{D}(H)$
are obtained. Each of those free (group) algebras is suitable for later applications of the results
in Section~\ref{sec:freegroupalgebrasdivision}. Thus the free generators (or elements obtained from them)
will be homogeneous elements of some graded rings that appear in this and subsequent sections.
There could be simpler elements that do the job and avoid some technicalities. But we were not able to find them.
Let $L$ be a nonabelian residually nilpotent Lie $k$-algebra over a field of characteristic zero $k$. The
main aim of Section~\ref{sec:residuallynilpotent} is to obtain free (group) algebras
in the division ring $\mathfrak{D}(L)$ from the ones obtained in the previous section. It is done by a method
involving series that was developed in \cite{FerreiraGoncalvesSanchez2}. Although technical,
the argument is quite natural.
Let $L$ be a nonabelian Lie $k$-algebra over a field of characteristic zero
such that its universal enveloping algebra $U(L)$ is an Ore domain. In Section~\ref{sec:Ore},
we find free group algebras in $\mathfrak{D}(L)$, the Ore ring of fractions of $U(L)$, using the results in
previous sections. Roughly speaking, the idea of the proof is that for some natural filtrations of $L$, the associated
graded Lie algebra $\gr(L)$ is residually nilpotent.
The isomorphism of graded algebras
$U(\gr(L))\cong \gr(U(L))$ allows us to use the results in previous sections
thanks to the fact that $U(L)$ is an Ore domain and the good behaviour of the Ore localization
with respect to filtrations described in Section~\ref{sec:filtrations&valuations}.
The arguments in Section~\ref{sec:Ore} should clarify why some of the elements in earlier sections where chosen
in that way. Here it is one of the places where Proposition~\ref{prop:freeobjecthomogeneous} and
Theorem~\ref{coro:divisionrings} are strongly used.
The last section of the paper is devoted to finding free group algebras in $k(G)$ for $k$ a field
of characteristic zero and $G$ a nonabelian residually torsion-free nilpotent group.
Let $\mathbb{H}=\langle a,b\colon (b,(b,a))=(a,(b,a))=1\rangle$ be
the Heisenberg group. There are filtrations of the group ring $k[\mathbb{H}]$ such
that the induced $k$-algebra is isomorphic to $U(H)$ as graded $k$-algebras, where
we consider a certain gradation in $U(H)$ induced from one of $H$. Again using the crucial
results of Section~\ref{sec:freegroupalgebrasdivision}, one can obtain suitable free group
algebras in $k(\mathbb{H})$. From this, using an argument from \cite{FerreiraGoncalvesSanchezFreegroupssymmetric}
one gets the desired free group algebras in $k(G)$.
\section{Filtrations, gradations and valuations}\label{sec:filtrations&valuations}
A \emph{strict ordering} on a set $S$ is a binary relation $<$ which is transitive and such that
$
|
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] | 1,024 |
ArXiv
| 0.674585 | 2.258789 | 0.091572 | 1.103985 |
s_1<s_2$ and $s_2<s_1$ cannot both hold for elements $s_1,s_2\in S$. It is a \emph{strict total
ordering} if for every $s_1,s_2\in S$ exactly one of $s_1<s_2$, $s_2<s_1$ or $s_1=s_2$ holds.
A group $G$ is called \emph{orderable} \label{orderablegroup} if its elements can be given a strict total ordering $<$
which is left and right invariant. That is, $g_1<g_2$ implies that $g_1h<g_2h$ and $hg_1<hg_2$ for all $g_1,g_2, h\in G$.
We call the pair $(G,<)$ an \emph{ordered group}.
Clearly, any additive subgroup of the real numbers is orderable. More generally, torsion-free abelian groups,
torsion-free nilpotent groups and residually torsion-free nilpotent groups are orderable \cite{Fuchs}.
We would like to point out that the results in this section are stated for ordered groups,
but in the following ones they will be applied for the ordered group $\mathbb{Z}$ alone.
We believe there is some merit in the general statements and they could be of interest to others.
\subsection{On filtrations and valuations}\label{sec:generalfiltrations}
Let $R$ be a ring and $(G,<)$ an ordered group. A family $F_GR=\{F_gR\}_{g\in G}$ of additive subgroups
of $R$ is a (descending)
\emph{$G$-filtration} if it satisfies the following four conditions
\begin{enumerate}[(F1)]
\item $F_gR\supseteq F_hR$ for all $g,h\in G$ with $g\leq h$;
\item $F_gR\cdot F_hR\subseteq F_{gh}R$ for all $g,h\in G$;
\item $1\in F_1R$;
\item $\bigcup\limits_{g\in G}F_gR=R$.
\end{enumerate}
We say that the $G$-filtration is \emph{separating} if it also satisfies
\begin{enumerate}[(F5)]
\item For every $x\in R$, there exists $g\in G$ such that $x\in F_gR$ and
$x\notin F_{h}R$ for all $h\in G$ with $g<h$.
\end{enumerate}
\medskip
Let $R$ be a ring, $(G,<)$ be an ordered group and $F_GR=\{F_gR\}_{g\in G}$ be a $G$-filtration of $R$.
For each $g\in G$, define $F_{>g}R=\sum_{h>g}F_hR$ and
$$R_g=F_gR/F_{>g}R.$$ The fact that $G$ is an ordered group and the definition of $G$-filtration imply that
$$F_{>g}R\cdot F_{>h}R\subseteq
F_{>gh}R,\
F_{>h}R\cdot F_{\geq g}R\subseteq F_{>gh}R,\ F_{\geq g}R\cdot F_{>h}R\subseteq
F_{>gh}R$$ for any
$g,h\in G$. Thus a multiplication can be defined by
\begin{equation}\label{eq:filtrationmultiplication}
R_g\times R_h\longrightarrow R_{gh},\quad
(x+F_{>g}R)(y+F_{>h}R)=xy+F_{>gh}R.
\end{equation}
The \emph{associated graded ring} of $F_GR$ is defined to
be
$$\gr_{F_G}(R)=\bigoplus_{g\in G} R_g.$$ The addition on $\gr_{F_G}(R)$ arises
from the addition on each component $R_g$. The multiplication is
defined by extending the multiplication
\eqref{eq:filtrationmultiplication}
|
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] | 1,024 |
ArXiv
| -0.240386 | 1.926556 | -0.689152 | 0.782706 |
on the components bilinearly to
all $\gr_{F_G}(R)$. Notice that $\gr_{F_G}(R)$ may not have
an identity element. If $F_GR$ is separating, then
$\gr_{F_G}(R)$ is a ring with identity element $1+F_{>1}R$.
The \emph{Rees ring} of the filtration is
$$\Rees_{F_G}(R)=\bigoplus_{g\in G}(F_gR) g,$$
which is a subring of the group ring $R[G]$.
Thus an element of $\Rees_{F_G}(R)$ is a finite sum
$\sum_{g\in G}a_gg$ where $a_g\in F_gR$.
Notice that $\Rees_{F_G}(R)$
is a $G$-graded ring with identity element $1_{\Rees_{F_G}(R)}=1_R1_G$.
The next lemma is well known. It can be proved as in \cite[Section~1.8]{MarubayashiVanOystaeyen} where
the filtrations are ascending.
\begin{lem}\label{lem:Reesgradedring}
Let $R$ be a ring, $(G,<)$ be an ordered group and
$F_GR=\{F_gR\}_{g\in G}$ be a $G$-filtration of $R$.
The following hold true.
\begin{enumerate}[\rm(1)]
\item The subset $G_1=\{g\in G\colon g\leq 1\}$ is an Ore subset of $\Rees_v(R)$ and
the Ore localization ${G_1}^{-1}\Rees_{F_G} (R)=R[G]$, the group ring.
\item Let $J$ be the ideal of $\Rees_{F_G}(R)$
generated by $G^-=\{g\in G\colon g<1\}$.
Then $J=\bigoplus\limits_{g\in G}(F_{>g}R)g$ and $\Rees_{F_G}(R)/J\cong \gr_{F_G}(R)$ as graded rings.
\item Let $I$ be the ideal of $\Rees_{F_G}(R)$ generated by the elements
$\{1-g\colon g\in G^-\}$. Then $\Rees_{F_G}(R)/I\cong R$. \qed
\end{enumerate}
\end{lem}
Let $R$ be a ring and $(G,<)$ be an ordered group. A map $\upsilon\colon R\rightarrow G\cup\{\infty\}$
is a \emph{valuation} if it satisfies
\begin{enumerate}[(V1)]
\item $\upsilon(x)=\infty$ if, and only if, $x=0$;
\item $\upsilon(x+y)\geq \min\{\upsilon(x),\upsilon(y)\}$;
\item $\upsilon(xy)=\upsilon(x)\upsilon(y)$.
\end{enumerate}
Notice that $\upsilon(1)=1_G$ and $\upsilon(-x)=\upsilon(x)$
for all $x\in R$.
For each $g\in G$, we set $R_{\geq g}=\{f\in R\colon \upsilon(f)\geq g\}$
and $R_{>g}=\{f\in R\colon \upsilon(f)>g\}$.
Defining $F_gR=R_{\geq g}$ for each $g\in G$,
we obtain a separating filtration $F_GR=\{F_gR\}_{g\in G}$. We will denote
the graded ring and the Rees ring associated to this filtration as
$\gr_\upsilon(R)$ and $\Rees_\upsilon(R)$, respectively. Furthermore,
observe that $\gr_\upsilon(R)$ is a domain because of (V3).
It is well known that the converse is also true \cite[p.91]{MarubayashiVanOystaeyen}.
That is, given a
separating filtration $F_GR=\{F_gR\}_{g\in G}$ of $R$ such that
the associated graded ring $\gr_{F_G}(R)$ is a domain, one can
define a valuation
$\
|
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] | 1,024 |
ArXiv
| 0.117842 | 2.212069 | -0.624372 | 0.711873 |
upsilon\colon R\rightarrow G\cup\{\infty\}$ by $\upsilon(x)=\max\{g\in G\colon x\in F_gR\}$
for each $x\in R\setminus\{0\}$.
If $X$ is an Ore domain, by $Q_{cl}(X)$, we denote the Ore ring of fractions of $X$
that is, the Ore localization of $X$ at the multiplicative set $X\setminus\{0\}$.
The following lemma is a generalization of
\cite[Propositions~16,17,18]{Lichtmanfreeuniversalenveloping}, with a
somewhat different proof.
\begin{lem}\label{lem:gradedOre}
Let $R$ be an Ore domain, $(G,<)$ be an ordered group and
$\upsilon\colon R\rightarrow G\cup\{\infty\}$ be a valuation.
Let $D$ be the Ore ring of fractions of $R$. The
following hold true.
\begin{enumerate}[\rm(1)]
\item The valuation $\upsilon$ can be extended to a valuation $\upsilon\colon D\rightarrow
G\cup\{\infty\}$.
\item The set $\mathcal{H}$ of nonzero homogeneous elements
of $\gr_\upsilon(R)$ is an Ore subset of $\gr_\upsilon(R)$.
\item There exists an isomorphism of $G$-graded rings
$\lambda\colon \mathcal{H}^{-1}\gr_\upsilon(R)\rightarrow \gr_\upsilon(D)$
given by $f+R_{>\upsilon(f)}\mapsto f+D_{>\upsilon(f)}$ for all
$f\in R$.
\item If $G$ is poly-(torsion-free abelian), then $\gr_\upsilon(R)$ is an Ore domain.
\item If $G$ is poly-(torsion-free abelian), then $\Rees_\upsilon(R)$ is an Ore domain.
\item If $G$ is torsion-free abelian, then $\mathcal{J}=\Rees_\upsilon(R)\setminus J$ is
an Ore subset of $\Rees_\upsilon(R)$ and $\mathcal{J}^{-1}\Rees_\upsilon(R)$ is a local ring with
residue division ring $Q_{cl}(\gr_\upsilon(R))$.
\end{enumerate}
\end{lem}
\begin{proof}
The proof of (1) can be found in \cite[Proposition~9.1.1]{Cohnskew}
for example.
(2) Let $f_1,f_2\in R\setminus\{0\}$. Consider the nonzero homogeneous elements
$f_1+R_{>\upsilon(f_1)}, f_2+R_{>\upsilon(f_2)}\in\gr_\upsilon
(R)$. Since $R$ is an Ore domain, there exist $q_1,q_2\in R$ such
that $q_1f_1=q_2f_2\neq 0$. Consider the nonzero homogeneous
elements $q_1+R_{>\upsilon(q_1)}, q_2+R_{>\upsilon(q_2)}\in\gr_\upsilon
(R)$. Then $$(q_1+R_{>\upsilon(q_1)})(f_1+R_{>\upsilon(f_1)})=
(q_2+R_{>\upsilon(q_2)})(f_2+R_{>\upsilon(f_2)}).$$
Now \cite[Lemma~8.1.1]{NastasescuvanOystaeyenMethodsgraded} implies
the result.
(3) First note that $\gr_\upsilon(D)$ is a $G$-graded skew field,
and the natural maps
$\iota\colon\gr_\upsilon(R)\hookrightarrow
\gr_\upsilon(D)$, $\kappa\colon \gr_\upsilon(R)\hookrightarrow
\mathcal{H}^{-1}\gr_\upsilon(R)$ are embeddings of $G$-graded rings. Thus, for each element
in $\mathcal{H}$, the image by $\iota$ is an homogeneous invertible
element in $\gr_\upsilon(D)$. By the universal property of the Ore
localization, there exists a homomorphism $\lambda\
|
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] | 1,024 |
ArXiv
| -0.366814 | 2.014806 | -0.823868 | 0.171771 |
colon
\mathcal{H}^{-1}\gr_\upsilon(R)\rightarrow \gr_\upsilon(D)$ such
that $\iota=\lambda\kappa$. The homomorphism $\lambda$ is injective
since it is so when restricted to homogeneous elements. Let now
$f,q\in R\setminus\{0\}$. Consider $q^{-1}f+D_{>\upsilon(q^{-1}f)}$. This
element is the image by $\lambda$ of
$(q+R_{>\upsilon(q)})^{-1}(f+R_{>\upsilon(f)})$. Thus $\lambda$ is
surjective.
(4) The graded division ring $\gr_\upsilon(D)$ is a crossed product
of the division ring $D_0$ over the subgroup $\{g\in G\colon D_g\neq 0\}$,
which is again poly-(torsion-free abelian). Thus $\gr_\upsilon(D)$ is an Ore domain by,
for example, \cite[Corollary~37.11]{Passman1}. We show that the Ore ring of fractions
$Q_{cl}(\gr_\upsilon(D))$ of $\gr_\upsilon(D)$ is also the Ore ring of fractions
of $\gr_\upsilon(R)$. For that, it is enough to show that every element of $Q_{cl}(\gr_\upsilon(D))$
is of the form $b^{-1}a$ with $a,b\in Q_{cl}(\gr_\upsilon(R))$, $b\neq 0$.
An element of $f\in Q_{cl}(\gr_\upsilon(D))$ is of the form
$(d_{g_1}+\dotsb +d_{g_r})^{-1}(e_{h_1}+\dotsb+e_{h_s})$ where
$d_{g_i}\in D_{g_i}$, $e_{h_j}\in D_{h_j}$. By (2),(3) and after bringing
to a common denominator, we may suppose that there exist
$t,a_i,b_j\in \mathcal{H}$ such that
$$f=(t^{-1}a_1+\dotsb+t^{-1}a_r)^{-1}(t^{-1}b_1+\dotsb+t^{-1}b_s)=
(a_1+\dotsb+a_r)^{-1}(b_1+\dotsb+b_s).$$
(5) In the same way as (4), one can show that the group ring
$D[G]$ and $R[G]$ are Ore domains with the same Ore ring of fractions $Q_{cl}(R[G])$.
By Lemma~\ref{lem:Reesgradedring}(1), $R[G]$ is the localization of
$\Rees_\upsilon(R)$ at $G_1$. Hence one can proceed as in (4) to show
that $\Rees_\upsilon(R)$ is an Ore domain with Ore ring of fractions $Q_{cl}(R[G])$.
(6) Let $x=\sum\limits_{i=1}^n a_ig_i\in\Rees_\upsilon(R)$ where
we suppose that $a_i\neq 0$, $i=1,\dotsc,n$. Hence $\upsilon(a_i)\geq g_i$ for all $i$.
We suppose that if $i<j$ either $\upsilon(a_i)^{-1}g_i<\upsilon(a_j)^{-1}g_j$ or
$\upsilon(a_i)^{-1}g_i=\upsilon(a_j)^{-1}g_j$ and $g_i<g_j$. We define
$\omega(x)=\upsilon(a_n)^{-1}g_n\leq 1_G$. Observe that
$x=x'\omega(x)$, where $$x'=\sum_{i=1}^n a_ig_ig_n^{-1}\upsilon(a_n).$$ Since
$\upsilon(a_n)^{-1}g_n\geq \upsilon(a_i)^{-1}g_i$, then
$g_i^{-1}\upsilon(a_i)\geq g_n^{-1}\upsilon(a_n)$. It implies that
$\upsilon(a_i)\geq g_ig_n^{-1}\upsilon(a_n)$.
Hence $x'\in \mathcal
|
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29918,
29876,
4935,
739,
10469,
393,
29871,
13,
4535,
29884,
3232,
29898,
29874,
29918,
29875,
2144,
6279,
330,
29918,
335,
29918,
29876,
3426,
29896,
1012,
29884,
3232,
29898,
29874,
29918,
29876,
4935,
13,
29950,
663,
395,
29916,
12764,
262,
320,
1942
] | 1,024 |
ArXiv
| -0.173818 | 2.139393 | -1.017536 | 0.155327 |
{J}$. Note that $x\in\mathcal{J}$ if and only if $\omega(x)=1$, since
$J=\bigoplus_{g\in G}F_{>g}R\cdot g$.
If $a,b\in R$, $g\in G$ such that $ag,bg\in\Rees_\upsilon(R)$, then
$$\omega((a+b)g)\leq
\max\{\omega(ag),\omega(bg)\}.$$ Let now
$y=\sum\limits_{j=1}^pb_jh_j\in \Rees_{\upsilon}(R)$
where we suppose that $b_j\neq 0$, $j=1,\dotsc,p$, and
if $j<l$ either $\upsilon(b_j)^{-1}h_j<\upsilon(b_l)^{-1}h_l$ or
$\upsilon(b_j)^{-1}h_j=\upsilon(b_l)^{-1}h_l$ and $h_j<h_l$.
Now note that $$xy=a_nb_pg_nh_p+\sum_{\{(i,j)\colon (i,j)\neq(n,p)\}}a_ib_jg_ih_j,$$
and if $(i,j)\neq(n,p)$, either $\upsilon(a_ib_j)^{-1}g_ih_j<\upsilon(a_nb_p)^{-1}g_nh_p$
or $g_ih_j<g_nh_p$. Therefore $\omega(xy)=\omega(x)\omega(y)$.
Let $u\in\mathcal{J}$ and $v\in\Rees_\upsilon(R)$.
Since $\Rees_\upsilon(R)$ is an Ore domain,
there exist $x,y\in \Rees_\upsilon(R)$ such that $xu=yv$. We have to prove that
$y$ can be chosen such that
$y\in\mathcal{J}$. From $xu=x'u\omega(x)=y'v'\omega(y)\omega(v)=yv$,
we get $x\omega(y)^{-1}u=y'v$, where $y'\in\mathcal{J}$.
Since $\omega(x)=\omega(y)\omega(v)$ with $\omega(v)\leq 1$,
then $\omega(x)\leq \omega(y)\leq 1$. It implies that
$x\omega(y)^{-1}\in\Rees_\upsilon(R)$ and $\omega(y')=1$.
\end{proof}
\subsection{On gradations and filtrations of universal enveloping algebras}\label{sec:filtrationuniversal}
If $L$ is a Lie algebra, we denote its
\emph{universal enveloping algebra} by $U(L)$.
Let $k$ be a field, $L$ be a Lie $k$-algebra and $G$ be a commutative group. We
say that $L$ is a \emph{$G$-graded Lie $k$-algebra} if there exists
a decomposition of $L$ as $L=\bigoplus\limits_{g\in G}L_g$ satisfying
\begin{enumerate}
\item $L_g$ is a $k$-subspace of $L$ for each $g\in G$,
\item $[L_g,L_h]\subseteq L_{g+h}$ for all $g,h\in G$.
\end{enumerate}
The elements of $\bigcup\limits_{g\in G} L_g$ are the
\emph{homogeneous elements} of $L$. If $x\in L_g$, we say that $x$
is \emph{homogeneous of degree} $g$.
The main examples we will deal with are the following.
Examples~(a),(b) are important
in Section~\ref{sec:nilpotentOre}, while examples (c),(d) are useful in Section~\ref{sec:Heisenberggroup}
\begin{ex}\label{ex:gradedLie}
Let $k$ be a field. We can endow the Heisenberg Lie $k$-algebra $H$
with different $\mathbb{Z}$-gradings. We will use the following ones.
\begin{enumerate}[(a)]
\item $H=\bigoplus_{
|
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29918,
29926,
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29913,
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29918,
4861,
29918,
29926,
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29884,
3232,
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9877,
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4935,
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13,
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262,
29905,
1942,
29912,
29967,
1042,
322,
395,
29894,
29905,
262,
29905,
1123,
267,
3187,
29884,
3232,
29898,
29934,
4935,
29871,
13,
4001,
779,
1123,
267,
3187,
29884,
3232,
29898,
29934,
1262,
338,
385,
438,
276,
5354,
29892,
13,
12711,
1863,
395,
29916,
29892,
29891,
29905,
262,
320,
1123,
267,
3187,
29884,
3232,
29898,
29934,
1262,
1316,
393,
395,
29916,
29884,
29922,
29891,
29894,
1504,
1334,
505,
304,
6356,
393,
13,
29938,
29891,
29938,
508,
367,
10434,
1316,
393,
13,
29938,
29891,
29905,
262,
29905,
1942,
29912,
29967,
4311,
3645,
395,
29916,
29884,
29922,
29916,
29915,
29884,
29905,
4787,
29898,
29916,
3892,
29891,
29915,
29894,
12764,
4787,
29898,
29891,
2144,
4787,
29898,
29894,
3892,
29891,
29894,
1628,
13,
705,
679,
395,
29916,
29905,
4787,
29898,
29891,
21604,
29896,
29913,
29884,
29922,
29891,
29915,
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29891,
2144,
4787,
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29894,
1262,
411,
779,
4787,
29898,
29894,
2144,
3797,
29871,
29896,
1628,
13,
6098,
779,
4787,
29898,
29916,
2144,
3797,
320,
4787,
29898,
29891,
2144,
3797,
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739,
10469,
393,
29871,
13,
29938,
29916,
29905,
4787,
29898,
29891,
21604,
29896,
1012,
262,
29905,
1123,
267,
3187,
29884,
3232,
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322,
779,
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29898,
29891,
1495,
29922,
29896,
1504,
13,
29905,
355,
29912,
8017,
29913,
13,
13,
13,
13,
13,
13,
13,
13,
13,
29905,
7235,
29912,
2951,
4656,
800,
322,
977,
509,
800,
310,
15968,
427,
1830,
292,
394,
28200,
1012,
1643,
29912,
3471,
29901,
1777,
509,
362,
14540,
284,
29913,
13,
13,
3644,
395,
29931,
29938,
338,
263,
7326,
9623,
29892,
591,
13530,
967,
13,
29905,
7278,
29912,
14540,
284,
427,
1830,
292,
9623,
29913,
491,
395,
29965,
29898,
29931,
4935,
13,
13,
13,
12024,
395,
29895,
29938,
367,
263,
1746,
29892,
395,
29931,
29938,
367,
263,
7326,
395,
29895,
4388,
15742,
322,
29871,
395,
29954,
29938,
367,
263,
26418,
1230,
2318,
29889,
1334,
13,
20834,
393,
395,
29931,
29938,
338,
263,
320,
7278,
8290,
29954,
4388,
5105,
287,
7326,
395,
29895,
4388,
15742,
29913,
565,
727,
4864,
13,
29874,
26227,
310,
395,
29931,
29938,
408,
395,
29931,
2013,
3752,
17201,
29905,
12514,
648,
29887,
29905,
262,
402,
29913,
29931,
29918,
29887,
29938,
24064,
13,
29905,
463,
29912,
15172,
29913,
13,
29905,
667,
395,
29931,
29918,
29887,
29938,
338,
263,
395,
29895,
4388,
1491,
3493,
310,
395,
29931,
29938,
363,
1269,
395,
29887,
29905,
262,
402,
1628,
13,
29905,
667,
11970,
29931,
29918,
29887,
29892,
29931,
29918,
29882,
10725,
11725,
365,
648,
29887,
29974,
29882,
1042,
363,
599,
395,
29887,
29892,
29882,
29905,
262,
402,
1504,
13,
29905,
355,
29912,
15172,
29913,
13,
1576,
3161,
310,
779,
3752,
5231,
29905,
12514,
648,
29887,
29905,
262,
402,
29913,
365,
29918,
29887,
29938,
526,
278,
13,
29905,
7278,
29912,
9706,
23724,
3161,
29913,
310,
395,
29931,
1504,
960,
395,
29916,
29905,
262,
365,
29918,
29887,
1628,
591,
1827,
393,
395,
29916,
29938,
13,
275,
320,
7278,
29912,
9706,
23724,
310,
7426,
29913,
395,
29887,
1504,
13,
13,
13,
1576,
1667,
6455,
591,
674,
5376,
411,
526,
278,
1494,
29889,
13,
1252,
9422,
30022,
29898,
29874,
21336,
29890,
29897,
526,
4100,
13,
262,
9779,
2651,
999,
29912,
3471,
29901,
8834,
17765,
296,
29949,
276,
1118,
1550,
6455,
313,
29883,
21336,
29881,
29897,
526,
5407,
297,
9779,
2651,
999,
29912,
3471,
29901,
3868,
7674,
2552,
2972,
29913,
13,
13,
29905,
463,
29912,
735,
1012,
1643,
29912,
735,
29901,
5105,
287,
29931,
347,
29913,
13,
12024,
395,
29895,
29938,
367,
263,
1746,
29889,
1334,
508,
1095,
340,
278,
940,
7674,
2552,
7326,
395,
29895,
4388,
15742,
395,
29950,
29938,
13,
2541,
1422,
779,
1995,
29912,
29999,
23021,
5105,
886,
29889,
1334,
674,
671,
278,
1494,
6743,
29889,
13,
29905,
463,
29912,
15172,
4400,
29898,
29874,
4638,
13,
29905,
667,
395,
29950,
2013,
3752,
17201,
648
] | 1,024 |
ArXiv
| -0.372163 | 1.85388 | -1.089265 | 0.131295 |
n\in\mathbb{Z}}H_n$, where $H_{-1}=kx+ky$, $H_{-2}=kz$ and $H_n=0$ for all $n\neq -1,-2.$
\item $H=\bigoplus_{n\in\mathbb{Z}}H_n$, where $H_{-1}=kx,$ $H_{-2}=ky$, $H_{-3}=kz$ and $H_n=0$ for all $n\neq -1,-2,-3$.
\item $H=\bigoplus_{n\in\mathbb{Z}}H_n$, where $H_{1}=kx+ky$, $H_{2}=kz$ and $H_n=0$ for all $n\neq 1,2.$
\item $H=\bigoplus_{n\in\mathbb{Z}}H_n$, where $H_{1}=kx,$ $H_{2}=ky$, $H_{3}=kz$ and $H_n=0$ for all $n\neq 1,2,3$.
\qed
\end{enumerate}
\end{ex}
For each $g\in G$, let $\mathcal{B}_g=\{e^g_i\}_{i\in I_g}$ be a
$k$-basis of $L_g$. Then $\mathcal{B}=\bigcup_{g\in G}\mathcal{B}_g$
is a $k$-basis of $L$. Fix an ordering $<$ of $\mathcal{B}$.
Consider the universal enveloping algebra $U(L)$ of $L$. The
\emph{standard monomials} in $\mathcal{B}$ are the elements
\begin{equation}\label{eq:standardmonomials}
e^{g_1}_{i_1}e^{g_2}_{i_2}\dotsb e^{g_r}_{i_r}\in U(L), \textrm{
with } e^{g_j}_{i_j}\in\mathcal{B}_{g_j},\ e^{g_1}_{i_1}\leq
e^{g_2}_{i_2}\leq \dotsb \leq e^{g_r}_{i_r}.
\end{equation}
By the Poincar\'e-Birkoff-Witt (PBW) theorem, the standard monomials, together with 1, form a
$k$-basis of $U(L)$. We say that the standard monomial
\eqref{eq:standardmonomials} is of degree $g=g_1+g_2+\dotsb+g_r.$
In this situation, one can obtain a gradation of the universal enveloping algebra as follows.
\begin{lem}\label{lem:gradeduniversalenveloping}
Let $G$ be a group and $L=\bigoplus\limits_{g\in G}L_g$ be a
$G$-graded Lie $k$-algebra. Then the universal enveloping algebra
$U(L)$ is an (associative) $G$-graded $k$-algebra. Indeed,
$$U(L)=\bigoplus_{g\in G} U(L)_g,$$ where
$U(L)_g=k\textrm{-span of
the standard monomials of degree }g$. \qed
\end{lem}
\bigskip
Let $k$ be a field, $L$ be a Lie $k$-algebra and $(G,<)$ be an ordered abelian group.
A (descending) \emph{separating filtration} of $L$ is a family of subspaces $F_GL=\{F_gL\}_{g\in G}$, such that
\begin{enumerate}[(FL1)]
\item $F_gL\supseteq F_hL$ for all $g,h\in G$ with $g\leq h$;
\item $[F_gL, F_hL]\subseteq F_{g+h}R$ for all $g,h\in G$;
\item $\bigcup_{g\in G}F_gL=L$;
\item For every $x\in L$, there exists $g\in G$ such that $x\in F_gL$ and
$x\notin F_{h}L$ for all $h\in G$ with $g<h$.
\end{enumerate}
Define $F_{>g}
|
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29916,
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395,
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648,
29906,
5369,
3459,
1628,
395,
29950,
648,
29941,
5369,
29895,
29920,
29938,
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] | 1,024 |
ArXiv
| -0.306274 | 2.071908 | -0.461189 | 0.498109 |
L=\sum_{h>g}F_hL$, and $L_g=F_gL/F_{>g}L$ for all $g\in G$. Then
one obtains the \emph{associated graded Lie $k$-algebra}
$$\gr_{F_G}{L}=\bigoplus_{g\in G} L_g.$$
The filtration $F_GL$ of $L$ induces a filtration $F_GU(L)=\{F_gU(L)\}_{g\in G}$
of the universal enveloping algebra
$U(L)$ as follows. Define, for each $g\in G$, $g\leq 0$,
$$F_gU(L)=k+\sum\limits_{g_1+\dotsb+g_r\geq g} L_{g_1}\dotsb L_{g_r},$$
and for each $g>0$
$$F_gU(L)=\sum_{g_1+\dotsb+g_r\geq g} L_{g_1}\dotsb L_{g_r}.$$
Then, $F_hU(L)\subseteq F_gU(L)$ for $g<h$, and $F_gU(L)
\cdot F_hU(L)\subseteq F_{g+h}U(L)$ for all $g,h\in G$.
An easy but important example for us is the following. It will be used in Section~\ref{sec:Ore}.
\begin{ex}\label{ex:usualfiltrationLiealgebra}
Let $L$ be a Lie $k$-algebra generated by two elements $u,v\in L$. Define $FL_r=0$ for all $r\geq 0$,
$FL_{-1}=ku+kv$, and, for $n\leq -1$,
$$F_{n-1}L=\sum_{n_1+n_2+\dotsb+n_r\geq (n-1)}[F_{n_1}L,[F_{n_2}L,\dotsc] \dotsb].$$
Observe that, for each $n\in\mathbb{Z}$, there exists $\mathcal{B}_{n}\subseteq L$
whose classes give a basis of $L_{n}=F_nL/F_{n+1}L$ such that
$\bigcup\limits_{n\in\integers} \mathcal{B}_{n}$ is a basis of $L$. \qed
\end{ex}
The next lemma will be used in Sections~\ref{sec:Ore},\ref{sec:Heisenberggroup}
\begin{lem}\label{lem:filtrationuniversalenveloping}
Let $k$ be a field and $L$ be a Lie $k$-algebra.
The following hold true.
\begin{enumerate}[\rm(1)]
\item Suppose that there exists a basis $\mathcal{B}_g=\{e^g_i\}_{i\in
I_g}$ of $L_g$ for each $g\in G$ such that $\bigcup\limits_{g\in
G}\mathcal{B}_g$ is a basis of $L$. Then the filtration is separating and
there exists an isomorphism of $G$-graded $k$-algebras
$$U(\gr_{F_G}(L))\cong\gr_{F_G}(U(L)).$$
Hence the filtration induces a valuation $\upsilon \colon U(L)\rightarrow G\cup\{\infty\}$.
\item If $U(L)$ is an Ore domain, then $U(\gr_{F_G}(L))$ is an
Ore domain.
\end{enumerate}
\end{lem}
\begin{proof}
(1) It can be proved in the same way as \cite[Proposition~1]{Vergne} or
\cite[Lemma~2.1.2]{Boiscorpsenveloppants}.
(2) By Lemma~\ref{lem:gradedOre}(4).
\end{proof}
\subsection{Free group algebras in division rings}\label{sec:freegroupalgebrasdivision}
Our work can be regarded as an application of some techniques on the
existence of free group algebras in division rings. In this section, we gather together
the version of those results that we will use.
We begin with \cite[
|
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] | 1,024 |
ArXiv
| 0.053397 | 2.097864 | -0.60375 | 0.544909 |
Theorem~3.2]{SanchezObtaininggraded}. It tells us
a way to obtain a free group algebra from a free algebra in case the division ring
is the Ore ring of fractions of a graded Ore domain.
\begin{theo}\label{theo:freegroupgradedOre}
Let $G$ be an orderable group and $k$ be a commutative ring. Let \linebreak
$A=\bigoplus\limits_{g\in G} A_g$ be a $G$-graded $k$-algebra.
Let $X$ be a subset of $A$
consisting of homogeneous elements where we denote by $g_x\in G$ the degree of $x\in X$,
i.e. $x\in A_{g_x}$. Suppose that the following three conditions are satisfied.
\begin{enumerate}[\rm(1)]
\item There exists a strict total ordering $<$ of $G$ such that $(G,<)$ is an ordered group and
$1<g_x$ for all $x\in X$.
\item The $k$-subalgebra of $A$ generated by $X$ is the free
$k$-algebra on $X$.
\item $A$ is a left Ore domain with left Ore ring of fractions $Q_{cl}(A)$.
\end{enumerate}
Then the $k$-subalgebra of $Q(A)$ generated by $\{1+x,\,
(1+x)^{-1}\}_{x\in X}$ is the free group $k$-algebra
on the set $\{1+x\}_{x\in X}$. \qed
\end{theo}
The next proposition is \cite[Proposition~2.5(4')]{SanchezObtainingI}.
It shows that the existence of a free group algebra in the graded ring
induced by a valuation on a division ring $D$, (under some circumstances)
implies the existence of a free group
algebra in $D$.
\begin{prop}\label{prop:freeobjecthomogeneous}
Let $Z$ be a commutative ring and $R$ be a $Z$-algebra. Let
$\upsilon\colon R\rightarrow \mathbb{Z}\cup\{\infty\}$ be a valuation. Let $X$
be a subset of elements of $R$ such that
the map $X\rightarrow \gr_\upsilon(R)$, $x\mapsto x+R_{>\upsilon(x)}$,
is injective.
\begin{enumerate}[\rm(1)]
\item The elements of $X$ are invertible in $R$.
\item The $Z_0$-subalgebra of $\gr_\upsilon(R)$ generated by
$\{x+R_{>\upsilon(x)},\, x^{-1}+R_{>\upsilon(x^{-1})}\}_{x\in X}$
is the free group $Z_0$-algebra on $\{x+R_{>\upsilon(x)}\}_{x\in
X}$.
\end{enumerate}
Then the $Z$-subalgebra of $R$ generated by $\{x,\,
x^{-1}\}_{x\in X}$ is the free group $Z$-algebra on $X$,
where $Z_0=Z_{\geq 0}/Z_{>0}$. \qed
\end{prop}
The next theorem is \cite[Theorem~3.2]{SanchezObtainingI}. It tells us that
sometimes, in order to find a free group algebra in division ring $D$,
it is enough to find a free algebra on the graded ring induced by
a valuation on $D$.
\begin{theo}\label{coro:divisionrings}
Let $D$ be a division ring with prime subring $Z$. Let $\upsilon\colon
D\rightarrow \mathbb{R}\cup\{\infty\}$ be a nontrivial valuation.
Let
$X$ be a subset of $D$ satisfying the following
three conditions.
\begin{enumerate}[\rm (1)]
\item The map $X\rightarrow\gr_\upsilon(D)$, $x\mapsto x+D_{>\upsilon(x)}$, is injective.
\item For each $x\in X$, $\upsilon(x)>0$.
\item The $Z_0$-subalgebra of $\gr_\upsilon(D)$ generated by the set
$\{x+D_{>\upsilon(x)}\}_{x\in X}$ is the
|
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] | 1,024 |
ArXiv
| 0.239176 | 2.279554 | -0.319076 | 1.093866 |
free $Z_0$-algebra
on the set $\{x+D_{>\upsilon(x)}\}_{x\in X}$, where
$Z_0= Z_{\geq 0}/Z_{>0}\subseteq D_0$.
\end{enumerate}
Then, for any central subfield $k$, the
$k$-subalgebra of $D$ generated by $\{1+x,\, (1+x)^{-1}\}_{x\in
X}$ is the free group $k$-algebra on $\{1+x\}_{x\in X}$. \qed
\end{theo}
\section{Nilpotent Lie algebras with involutions}\label{sec:nilpotentLieinvolutions}
Let $k$ be a field and $L$ be a
Lie $k$-algebra. A $k$-linear map $*\colon L\rightarrow L$ is a
$k$-\emph{involution} \label{kinvolutionLie}
if for all $x,y\in L$, $[x,y]^*=[y^*, x^*]$,
$x^{**}= x$. The main example of a $k$-involution in a Lie
$k$-algebra is what we call the \emph{principal involution}. It is
defined by $x\mapsto -x$ for all $x\in L$.
The \emph{Heisenberg Lie $k$-algebra} is the Lie $k$-algebra with
presentation
\begin{equation}\label{eq:Heisenberg}
H=\langle x,y\mid [[y,x],x]=[[y,x],y]=0 \rangle.
\end{equation}
The Heisenberg Lie $k$-algebra can also be characterized as the
unique Lie $k$-algebra of dimension three such that $[H,H]$ has
dimension one and $[H,H]$ is contained in the center of $H$, see
\cite[Section~4.III]{JacobsonLiealgebras}.
Let $k$ be a field of characteristic different from $2$.
In this section, we first find all the $k$-involutions of $H$,
secondly we show that there are essentially three involutions on $H$, and then
that any nilpotent Lie $k$-algebra with involution
contains a $k$-subalgebra isomorphic to $H$ invariant under the involution and such
that the restriction of the involution to $H$ is one of those three ones.
\begin{lem}\label{lem:involutionHeisenbergalgebra}
Let $k$ be a field of characteristic different from two. Let \linebreak $H=\langle x,y \mid
[x,[y,x]]=[y,[y,x]]=0 \rangle$ be the Heisenberg Lie $k$-algebra and $z=[y,x]$.
Then any $k$-involution $\tau\colon H\rightarrow H$ is of one of
the following forms:
\begin{enumerate}[\rm(i)]
\item
$\left\{\begin{array}{l} \tau(x)= ax+by+cz \\
\tau(y)= dx-ay+fz \\ \tau(z)=z \end{array}\right.$ where $a,b,c,d,f\in k$
satisfy \ $\left\{\begin{array}{rrr}
a^2 & + & bd=1, \\
(a+1)c & + & bf=0 \\ dc & +& (1-a)f=0
\end{array}\right. $
\item $\left\{\begin{array}{l} \tau(x)= x+cz \\
\tau(y)= y+fz \\ \tau(z)= -z \end{array}\right.$ where $c,f\in k$.
\item $\left\{\begin{array}{l} \tau(x)= -x \\
\tau(y)= -y \\ \tau(z)= -z \end{array}\right.$
\end{enumerate}
\end{lem}
\begin{proof}
Let $*\colon H\rightarrow H$ be a $k$-involution on $H$. Note that
$Z(H)$, the center of $H$, is the one-dimensional $k$-subspace
generated by $z=[y,x]$. Since $z^*\in Z(H)$ and $(z^*)^
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] | 1,024 |
ArXiv
| -0.338964 | 2.030379 | -0.740035 | 0.212247 |
*=z$, we
obtain that $z^*=z$ or $z^*=-z$.
Suppose that $x^*=ax+by+cz$ and $y^*=dx+ey+fz$ where $a,b,c,d,e,f\in
k$.
\noindent{\underline{Case 1: $z^*=z$}.}
Then
$z=[y,x]=[y,x]^*=[x^*,y^*]=[ax+by+cz,
dx+ey+fz]=[by,dx]+[ax,ey]=(bd-ae)z$. Thus
\begin{equation}\label{case1_*} bd-ae=1
\end{equation}
From
$x=(x^*)^*=(ax+by+cz)^*=a(ax+by+cz)+b(dx+ey+fz)+cz=(a^2+bd)x+(ab+be)y+(ac+bf+c)z$,
we get
\begin{eqnarray}
& a^2+bd=1, & \label{case1_a}\\
& b(a+e)=0, & \label{case1_b}\\
& ac+bf+c=0. & \label{case1_c}
\end{eqnarray}
From
$y=(y^*)^*=(dx+ey+fz)^*=d(ax+by+cz)+e(dx+ey+fz)+fz=d(a+e)x+(e^2+db)y+(cd+ef+f)z$,
we obtain
\begin{eqnarray}
& e^2+bd=1, & \label{case1_d}\\
& d(a+e)=0, & \label{case1_e}\\
& cd+ef+f=0. & \label{case1_f}
\end{eqnarray}
From \eqref{case1_a} and \eqref{case1_d}, we obtain that $a=\pm e$.
Suppose that $a=e$. Then \eqref{case1_a} and \eqref{case1_*} imply
that $a=e=0$. Thus. this case is contained in the case $a=-e$.
Suppose now that $a=-e$. Then \eqref{case1_a}, \eqref{case1_d} and
\eqref{case1_*} are in fact the same equation. Also the equations
\eqref{case1_b} and \eqref{case1_e} do not give any new information.
Thus \eqref{case1_c} and \eqref{case1_f} equal
\begin{eqnarray}\label{case1_coef}
\left\{ \begin{array}{rrr} (a+1)c & + & bf=0 \\ dc & +& (1-a)f=0 \end{array}\right.
\end{eqnarray}
Observe that, by \eqref{case1_a}, $\det \begin{pmatrix} a+1 & b \\ d
& 1-a\end{pmatrix}=-a^2-bd+1=0$.
Therefore $x^*=ax+by+cz$, $y^*=dx-ay+fz$, $z^*=z$, where $a,b,c,d,f$
satisfy \eqref{case1_a} and \eqref{case1_coef}. Hence (i) is proved.
\noindent{\underline{Case 2: $z^*=-z$}.}
$-z=[y,x]^*=[x^*,y^*]=[ax+by+cz,
dx+ey+fz]=[by,dx]+[ax,ey]=(bd-ae)z$. Thus
\begin{equation}\label{case2_*} ae-bd=1
\end{equation}
From
$x=(x^*)^*=(ax+by+cz)^*=a(ax+by+cz)+b(dx+ey+fz)-cz=(a^2+bd)x+(ab+eb)y+(ac+bf-c)z$,
we get
\begin{eqnarray}
& a^2+bd=1, & \label{case2_a}\\
& b(a+e)=0, & \label{case2_b}\\
& ac+bf-c=0. & \label{case2_c}
\end{eqnarray}
|
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] | 1,024 |
ArXiv
| -0.664927 | 1.308811 | -1.361835 | -0.144449 |
From
$y=(y^*)^*=(dx+ey+fz)^*=d(ax+by+cz)+e(dx+ey+fz)-fz=d(a+e)x+(e^2+db)y+(cd+ef-f)z$,
we obtain
\begin{eqnarray}
& e^2+bd=1, & \label{case2_d}\\
& d(a+e)=0, & \label{case2_e}\\
& cd+ef-f=0. & \label{case2_f}
\end{eqnarray}
From \eqref{case2_a} and \eqref{case2_d}, we obtain that $a=\pm e$.
It is not possible that $a=-e$ because \eqref{case2_a} and
\eqref{case2_*} would imply that $1=-1$.
Suppose now that $a=e$. Then \eqref{case2_*}, \eqref{case2_a} and
\eqref{case2_d} imply that $a^2=1$. Hence $a=e=\pm 1$. Now
\eqref{case2_b} and \eqref{case2_e} imply that $b=d=0$.
If $a=-1$, we obtain, by \eqref{case2_c} and \eqref{case2_f}, that
$f=c=0$. Hence we obtain (iii), i.e. $x^*=-x$, $y^*=-y$ $z^*=-z$.
If $a=1$, \eqref{case2_c} and \eqref{case2_f} do not give any new
information. Hence we obtain (ii), i.e. $x^*=x+cz$, $y^*=y+fz$,
$z^*=-z$, where $c,f\in k$.
\end{proof}
Let $k$ be a field.
Let $\tau,\eta\colon L\rightarrow L$ be two $k$-involutions of a Lie $k$-algebra $L$.
We say that $\tau$ is \emph{equivalent} to $\eta$ if there exists an isomophism of Lie $k$-algebras
$\varphi\colon L\rightarrow L$ such that $\varphi^{-1}\tau\varphi=\eta$.
\begin{lem}\label{lem:equivalentinvolutionHeisenbergalgebra}
Let $k$ be a field of characteristic different from two and $H$ be the Heisenberg Lie $k$-algebra. Any
$k$-involution $\tau\colon H\rightarrow H$ is equivalent to one of the following involutions
$\eta\colon H\rightarrow H$.
\begin{enumerate}[\rm (1)]
\item The involution
$\eta\colon H\rightarrow H$ defined by $\eta(x)=x,\, \eta(y)= -y,\, \eta(z)= z.$
More precisely, any $k$-involution in
Lemma~\ref{lem:involutionHeisenbergalgebra}{\rm(i)} is equivalent to $\eta$ just defined.
\item The involution $\eta\colon H\rightarrow H$ defined by $\eta(x)= x,\, \eta(y)= y,\, \eta(z)= -z$.
More precisely, any $k$-involution in Lemma~\ref{lem:involutionHeisenbergalgebra}{\rm(ii)}
is equivalent to $\eta$ just defined.
\item The principal involution $\eta\colon H\rightarrow H$ defined by $\eta(x)=-x,\, \eta(y)=-y,\, \eta(z)=-z$.
\end{enumerate}
Furthermore, we exhibit explicit isomorphisms $\varphi\colon
H\rightarrow H$ which prove that $\varphi^{-1}\tau\varphi=\eta$ where $\tau$ is any involution in
Lemma~\ref{lem:involutionHeisenbergalgebra}{\rm(i)} and {\rm(ii)}.
\end{lem}
\begin{proof}
Clearly the involution of Lemma~\ref{lem:involutionHeisenbergalgebra}(iii) is the same as the one in (3).
First we prove (2). Let $f\mapsto f^*$ be any involution in Lemma~\ref{lem:involutionHeisenbergalgebra}(ii).
Suppose that $c,f\in k$
|
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] | 1,024 |
ArXiv
| -0.278085 | 1.85388 | -1.037262 | 0.250193 |
and that $x^*=x+cz$,
$y^*=y+fz$ and $z^*=-z$. Define $X=\frac{1}{2}(x+x^*)=\frac{1}{2}(2x+cz)$,
$Y=\frac{1}{2}(y+y^*)=\frac{1}{2}(2y+fz)$ and $Z=z$.
Note that $X,Y, Z$ form a $k$-basis of $H$ and that $[Y,X]=[y,x]=z=Z$. Thus
there exists an isomorphism $\varphi\colon H\rightarrow H$ sending $x\mapsto X$, $y\mapsto Y$ and
$z\mapsto Z$. Moreover $X^*=X$, $Y^*=Y$ and $Z^*=-Z$, as desired.
Now we prove (1). Let $h\mapsto h^*$ be any involution of
Lemma~\ref{lem:involutionHeisenbergalgebra}(i).
Let $a,b,c,d,f\in k$ satisfying the
conditions in Lemma~\ref{lem:involutionHeisenbergalgebra}(i). Hence $x^*=ax+by+cz $,
$y^*=dx-ay+fz$, $z^*=z$. We consider three cases:
\begin{enumerate}[(I)]
\item $b\neq 0$
\item $d\neq 0$
\item $b=d=0$.
\end{enumerate}
(I) Suppose $b\neq 0$. Define $X=\frac{1+a}{2}x+\frac{b}{2}y+\frac{c}{2}z$, $Y=\frac{1-a}{2}x-\frac{b}{2}y-\frac{c}{2}z$ and $Z=-\frac{b}{2}z$.
Note that $X,Y,Z$ is a $k$-basis of $H$ and that $[Y,X]=Z.$ Thus
there exists an isomorphism of $\varphi\colon H\rightarrow H$ sending $x\mapsto X$, $y\mapsto
Y$, $z\mapsto Z$. Note that $X^*=X$, $Y^*=-Y$ and $Z^*=Z$, as
desired.
(II) Suppose now that $d\neq 0$. Define $X=\frac{d}{2}x+\frac{1-a}{2}y+\frac{f}{2}z$, $Y=\frac{d}{2}x-\frac{a+1}{2}y+\frac{f}{2}z$ and
$Z=-\frac{d}{2}z$. Note that $X,Y,Z$ is a $k$-basis of $H$ and that $[Y,X]=Z$.
Thus there exists an isomorphism $\varphi\colon H\rightarrow H$ given by $x\mapsto X$,
$y\mapsto Y$, $z\mapsto Z$. Note that $X^*=X$, $Y^*=-Y$ and $Z^*=Z$,
as desired.
(III) Suppose that $b=d=0$. Then $a^2=1$ and either $c=0$ or $f=0$.
In both cases define $X=-\frac{1+a}{2}x+\frac{1-a}{2}y+\frac{f-c}{2}z$, $Y=\frac{a-1}{2}x+\frac{1+a}{2}y+\frac{c-f}{2}z$, $Z=-az$. It
is not difficult to show that $X,Y,Z$ is a $k$-basis of $H$ and that
$[Y,X]=Z$. Thus there exists an isomorphism $\varphi\colon H\rightarrow H$ given by
$x\mapsto X$, $y\mapsto Y$, $z\mapsto Z$. Note that $X^*=X$, $Y^*=-Y$
and $Z^*=Z$, as desired.
\end{proof}
The following two results are the Lie algebra version of
\cite[Lemma~2.3, Proposition~2.4]{FerreiraGoncalvesSanchezFreegroupssymmetric}.
The proofs are analogous to the ones given there for groups.
\begin{lem}\label{lem:involutionclass2}
Let $k$ be a field of characteristic different from two, and $L$ be a finitely generated nilpotent
Lie
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] | 1,024 |
ArXiv
| 0.108757 | 1.910983 | -0.800556 | 0.486725 |
$k$-algebra of class $2$ with involution $*\colon L\rightarrow L$, $f\mapsto f^*$.
Then $L$ contains a $*$-invariant Heisenberg Lie $k$-subalgebra $H$ and the
restriction to $H$ is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}. More precisely,
there exist $x,y\in L$ such that $[y,x]\neq0$,
$[y,[y,x]]=[x,[y,x]]=0$, and either
$x^*=x,\ y^*=-y$, or $x^*=x,\ y^*=y$, or $x^*=-x,\ y^*=-y$.
\end{lem}
\begin{proof}
Let $C$ denote the center of $L$. It follows from the nilpotency class of $L$
that $L/C$ is a finitely generated torsion-free abelian Lie $k$-algebra and the involution
induces an automorphism of $k$-vector spaces $\varphi\colon L/C\rightarrow L/C$, $f+C\mapsto f^*+C$.
Notice that $L/C$ has dimension at least two because $L$ is not abelian. Since $\varphi^2$ is
the identity, $\varphi$ is diagonalizable. There exist $u_1,\dotsc,u_n\in L$ such
that $\{u_1+C,\dotsc,u_n+C\}$ is a basis of $L/C$ consisting of eigenvectors with eigenvalues
$\pm 1$. Since $L$ is not abelian, we may suppose that $[u_1,u_2]\neq 0$.
Hence there exist $z_1,z_2\in C$ such that $u_i^*=\varepsilon_iu_i+z_i$,
where $\varepsilon_i\in\{1,-1\}$, for $i=1,2$.
Suppose that $u_1^*=-u_1+z_1$ and $u_2^*=u_2+z_2$.
Let $H$ be the subalgebra with basis $x=\frac{1}{2}(u_1- u_1^*)=u_1-\frac{1}{2}z_1$,
$y=\frac{1}{2}(u_2+u_2^*)=u_2+\frac{1}{2}z_2$ and $z=[y,x]$. Now proceed as in the previous case.
Suppose that $u_1^*=u_1+z_1$ and $u_2^*=-u_2+z_2$.
Let $H$ be the subalgebra with basis $x=\frac{1}{2}(u_2- u_2^*)=u_2-\frac{1}{2}z_2$,
$y=\frac{1}{2}(u_1+u_1^*)=u_2+\frac{1}{2}z_2$ and $z=[y,x]$.
Clearly $z$ commutes with $x$ and $y$ because it is an element of $C$.
Now $x^*=-x$, $y^*=y$ and
$z^*=[y,x]^*=[x^*,y^*]=[-x,y]=z$. Thus
$*$, when restricted to $H$, is the involution (1) in Lemma~\ref{lem:involutionHeisenbergalgebra}.
Suppose that $u_1^*=u_1+z_1$ and $u_2^*=u_2+z_2$. Let $H$
be the subalgebra with basis $x=\frac{1}{2}(u_1+ u_1^*)=u_1+\frac{1}{2}z_1$,
$y=\frac{1}{2}(u_2+u_2^*)=u_2+\frac{1}{2}z_2$ and $z=[y,x]$.
Clearly $z$ commutes with $x$ and $y$ because it is an element of $C$. Now $x^*=x$, $y^*=y$ and
$z^*=[y,x]^*=[x^*,y^*]=[x,y]=-z$.
Thus $*$, when restricted to $H$, is the involution (2) in Lemma~\ref{lem:involutionHeisenbergalgebra}.
Suppose
|
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] | 1,024 |
ArXiv
| -0.306954 | 1.931747 | -1.117957 | 0.069316 |
that $u_1^*=-u_1+z_1$ and $u_2^*=-u_2+z_2$. Let $H$
be the subalgebra with basis $x=\frac{1}{2}(u_1- u_1^*)=u_1-\frac{1}{2}z_1$,
$y=\frac{1}{2}(u_2-u_2^*)=u_2-\frac{1}{2}z_2$ and $z=[y,x]$. Clearly $z$ commutes with $x$ and $y$ because it is an element of $C$. Now $x^*=-x$, $y^*=-y$ and
$z^*=[y,x]^*=[x^*,y^*]=[-x,-y]=-z$. Thus $*$,
when restricted to $H$, is the involution (3) in Lemma~\ref{lem:involutionHeisenbergalgebra}
\end{proof}
\begin{prop}\label{prop:involutionnilpotent}
Let $k$ be a field of characteristic different from two, and $L$ be a non-abelian nilpotent
Lie $k$-algebra with involution $*\colon L\rightarrow L$, $f\mapsto f^*$.
Then $L$ contains a $*$-invariant Heisenberg Lie $k$-subalgebra $H$ such
that the restriction to $H$ is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}. More precisely,
there exist $x,y\in L$ such that $[y,x]\neq0$,
$[y,[y,x]]=[x,[y,x]]=0$, and either
$x^*=x,\ y^*=-y$, or $x^*=x,\ y^*=y$, or $x^*=-x,\ y^*=-y$.
\end{prop}
\begin{proof}
By taking the subalgebra of $L$ generated by two noncommuting elements and their
images by $*$, we can assume that $L$ is finitely generated.
We shall argue by induction on the nilpotency class $c$ of $L$; the case $c=2$
having been dealt with in Lemma~\ref{lem:involutionclass2}.
Suppose that $c>2$ and let $C$ denote the center of $L$. Then $L/C$
is a nonabelian finitely generated nilpotent Lie algebra of class $c-1$, with
an involution induced by $*$. By the induction hypothesis, there exist $x,y\in L$
such that $\{x+C,y+C\}$ generate a $*$-invariant Heisenberg Lie subalgebra of $L/C$.
Moreover $x^*+C=\varepsilon x+C$ and $y^*+C=\eta y+C$ with $\varepsilon,\eta\in\{1,-1\}$
and $z=[y,x]\notin C$, $[y,z],[x,z]\in C$. It follows that $M$, the
subalgebra of $L$ generated by $\{x,y,C\}$, is a $*$-invariant subalgebra of $L$
of nilpotency class $\leq3$.
If $M$ has class 2, then the result follows from Lemma~\ref{lem:involutionclass2}.
Suppose that $M$ has class 3. Then $[x,z]\neq 0$ or $[y,z]\neq 0$. Say $[x,z]\neq 0$.
We shall show that the $k$-subalgebra $K$ generated by $\{x,x^*,z,z^*\}$ is a $*$-invariant
subalgebra of $L$ of class $2$. It will be enough to show that $[\alpha,\beta]$
lies in the center of $K$ for all $\alpha,\beta\in \{x,x^*,z,z^*\}$. For each $n\geq 1$,
let $\gamma_n(M)$ denote the $n$-th term in the lower central series of $M$. Now,
$z=[y,x]\in\gamma_2(M)$, so $z^*\in\gamma_2(M)$, because the terms in the lower central
series are fully invariant subgroups of $G$. It follows that for every $\alpha\in K$ and
$\beta\in \{z,z^*\}$ we haver $[\alpha,\beta]\in\gamma_3(M)$, which is a
|
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] | 1,024 |
ArXiv
| 0.131088 | 2.248407 | -0.485846 | 0.678986 |
central subalgebra
of $M$, hence $[\alpha,\beta]$ is central in $K$. Finally, that $[x,x^*]=0$ follows
from the fact that $x^*-x\in C$. So $K$ is indeed a $*$-invariant subalgebra of $L$ of class $2$.
Hence Lemma~\ref{lem:involutionclass2} applies to it.
\end{proof}
\section{Free group algebras in the Ore ring of fractions of universal enveloping algebras
of nilpotent Lie algebras}\label{sec:nilpotentOre}
Let $k$ be a field of characteristic zero and $H$ be the Heisenberg Lie $k$-algebra.
In this section we find different free (group) $k$-subalgebras in $\mathfrak{D}(H)$,
the Ore ring of fraction of the universal enveloping algebra $U(H)$ of $H$.
For that, our main tool is the result by G. Cauchon~\cite[Th\'eor\`eme]{Cauchoncorps}.
The technique to obtain suitable free algebras from the paper of Cauchon was developed in
\cite[Section~3]{FerreiraGoncalvesSanchez2}. In some cases, we then use Theorem~\ref{theo:freegroupgradedOre}
to obtain free group algebras in $\mathfrak{D}(H)$. If $L$ is a nilpotent Lie $k$-algebra,
applying Proposition~\ref{prop:involutionnilpotent},
the foregoing implies the existence of free group algebras in $\mathfrak{D}(L)$, the Ore ring of fractions
of the universal enveloping algebra $U(L)$ of $L$.
\medskip
Let $k$ be a field of characteristic different from two. Let $K=k(t)$ be the field of fractions of the
polynomial ring $k[t]$ in the variable $t$. Let $\sigma$ be a
$k$-automorphism of $K$ of \emph{infinite order}. We will consider the skew polynomial ring
$K[p;\sigma]$. The elements of $K[p;\sigma]$ are ``right
polynomials'' of the form $\sum_{i=0}^n p^ia_i$, where the
coeficients $a_i$ are in $K$. The multiplication is determined by
$$ap=p\sigma(a) \quad \textrm{for all } a\in K.$$
It is known that $K[p;\sigma]$ is a noetherian domain and therefore
it has an Ore division ring of fractions $D=K(p;\sigma)$.
Since $\sigma$ is an automorphism of $K$,
$\sigma(t)=\frac{at+b}{ct+d}$ where $M=\left(\begin{smallmatrix} a & b \\
c & d
\end{smallmatrix}\right)\in GL_2(k)$ defines a homography $h$ of the
projective line $\Delta=\mathbb{P}_1(k)=k\cup\{\infty\}$, $h\colon
\Delta \rightarrow \Delta,$ $z\mapsto h(z)=\frac{az+b}{cz+d}$.
We denote by $\mathcal{H}=\{h^n\mid n\in\mathbb{Z} \}$ the subgroup
of the projective linear group $PGL_2(k)$ generated by $h$. The
group $\mathcal{H}$ acts on $\Delta$. If $z\in\Delta$, we denote by
$\mathcal{H}\cdot z=\{h^n(z)\mid n\in\mathbb{Z}\}$ the orbit of $z$
under the action of $\mathcal{H}$.
\begin{theo}[Cauchon's Theorem]
Let $\alpha$ and $\beta$ be two elements of $k$ such that the orbits
$\mathcal{H}\cdot \alpha$ and $\mathcal{H}\cdot \beta$ are infinite
and different. Let $s$ and $u$ be the two elements of $D$ defined by
$$s=(t-\alpha)(t-\beta)^{-1}, \quad u=(1-p)(1+p)^{-1}.$$
If the characteristic of $k$ is different from $2$, then the
$k$-subalgebra $\Omega$ of $D$ generated by $\xi=s$,
$\eta=usu^{-1
|
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] | 1,024 |
ArXiv
| 0.208407 | 2.129011 | -0.536056 | 0.471546 |
}$, $\xi^{-1}$ and $\eta^{-1}$ is the free group
$k$-algebra on the set $\{\xi,\eta\}$. \qed
\end{theo}
We will need the following consequence of Cauchon's Theorem.
\begin{prop}\label{prop:freealgebrainWeyl}
Let $k$ be a field of characteristic zero and $K=k(t)$ be the field of
fractions of the polynomial ring $k[t]$. Let $\sigma\colon
K\rightarrow K$ be the automorphism of $k$-algebras determined by
$\sigma(t)=t-1$. Consider the skew polynomial ring $K[p;\sigma]$ and
its Ore division ring of fractions $K(p;\sigma)$.
Set $s=(t-\frac{5}{6})(t-\frac{1}{6})^{-1}$, $u=(1-p^2)(1+p^2)^{-1}$ and
$u_1=(1-p^3)(1+p^3)^{-1}$. The following hold true.
\begin{enumerate}[\rm(1)]
\item The
$k$-subalgebra of $K(p;\sigma)$ generated by $\{s,\, s^{-1},\ usu^{-1},\, us^{-1}u^{-1}\}$ is the free
group $k$-algebra on the set
$\{s,\ usu^{-1}\}$.
\item The $k$-subalgebra of $K(p;\sigma)$ generated by $\{s+s^{-1},\
u(s+s^{-1})u^{-1}\}$ is the free $k$-algebra on the set
$\{s+s^{-1},\ u(s+s^{-1})u^{-1}\}$.
\item The $k$-subalgebra of $K(p;\sigma)$ generated by $\{s+s^{-1},\
u_1(s+s^{-1})u_1^{-1}\}$ is the free $k$-algebra on the set
$\{s+s^{-1},\ u_1(s+s^{-1})u_1^{-1}\}$.
\end{enumerate}
\end{prop}
\begin{proof}
We will apply Cauchon's Theorem to the skew polynomial ring
$K[p^2;\sigma^2]$, where $\sigma^2\colon K\rightarrow K$ is given by
$\sigma^2(t)=t-2$.
Let $\alpha=\frac{5}{6}\in k$ and $\beta=\frac{1}{6}$. Let
$\mathcal{H}$ be defined as above. Consider the orbits
$\mathcal{H}\cdot\alpha=\{\frac{5}{6}-2n\mid n\in\mathbb{Z}\}$,
$\mathcal{H}\cdot\beta=\{\frac{1}{6}-2n\mid n\in\mathbb{Z}\}$ which
are infinite and different.
Then, by Cauchon's Theorem, $s=(t-\alpha)(t-\beta)^{-1}$ and
$u=(1-p^2)(1+p^2)^{-1}$ are such that the $k$-algebra generated by
$\xi=s, \eta=usu^{-1}$, $\xi^{-1}$ and $\eta^{-1}$ is the free group
$k$-algebra on the free generators $\{\xi,\eta\}$. Thus (1) is proved.
By Corollary~\ref{coro:freeinsidegroupring}, the $k$-algebra
generated by $\{s+s^{-1},\ u(s+s^{-1})u^{-1}\}$ is the free
$k$-algebra on the set $\{s+s^{-1},\ u(s+s^{-1})u^{-1}\}$. Thus (2) is proved.
In order to prove (3), apply Cauchon's Theorem to the skew polynomial ring
$K[p^3;\sigma^3]$, where $\sigma^3\colon K\rightarrow K$ is given by
$\sigma^3(t)=t-3$. Then proceed as in (1) and (2).
\end{proof}
The following lemma is well known. For example, it appears in \cite[Section~3.3]{FerreiraGoncalvesSanchez2}.
\begin{lem}\label{lem:specializationtoWeyl}
Let $k$ be a field of characteristic zero. Let $K=k(t)$ be the field of fractions of the
|
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] | 1,024 |
ArXiv
| 0.33155 | 2.154966 | -0.238381 | 0.810533 |
polynomial
ring $k[t]$ and $\sigma\colon K\rightarrow K$ be the automorphism of $k$-algebras determined by
$\sigma(t)=t-1$. Consider the skew polynomial ring $K[p;\sigma]$ and its Ore ring of fractions
$K(p;\sigma)$. Let $H=\langle x,y\mid [[y,x],x]=[[y,x],y]=0 \rangle$ be the Heisenberg Lie $k$-algebra, set $z=[y,x]$
and consider the universal enveloping algebra $U(H)$ of $H$.
. The following hold true
\begin{enumerate}[\rm(1)]
\item Set $I=U(H)(z-1)$, the ideal of $U(H)$
generated by $z-1$. The set $\mathfrak{S}=U(H)\setminus I$ is a left Ore subset of $U(H)$.
\item There exists a surjective $k$-algebra homomorphism $\Phi\colon \mathfrak{S}^{-1}U(H)\rightarrow K(p;\sigma)$
such that $\Phi(y)=p$, $\Phi(x)=p^{-1}t$ and $\Phi(z)=1$.
\end{enumerate}
\end{lem}
\begin{proof}
First note that $$p(p^{-1}t)-(p^{-1}t)p=t-p^{-1}p(t-1)=1.$$
Hence there exists a $k$-algebra homomorphism $\Phi\colon U(H)\rightarrow K(p;\sigma)$
such that $\Phi(y)=p$, $\Phi(x)=p^{-1}t$ and $\Phi(z)=1$. The ideal $I$ is clearly contained in the kernel of $\Phi$.
Now note that $U(H)/I$ is the first Weyl algebra, which is a simple $k$-algebra. Thus $I$ is the kernel of $\Phi$.
The subset $\mathfrak{S}$ is an Ore subset of $U(H)$ by \cite[Lemma~13]{Lichtmanfreeuniversalenveloping}.
By the universal property of the Ore localization $\Phi$ can be extended to a $k$-algebra
homomorphism $\Phi\colon \mathfrak{S}^{-1}U(H)\rightarrow K(p;\sigma)$. Note that $\mathfrak{S}^{-1}U(H)$
is a local ring with maximal ideal $\mathfrak{S}^{-1}I$. It induces an embedding of
division rings $\mathfrak{S}^{-1}U(H)/\mathfrak{S}^{-1}I\rightarrow K(p;\sigma)$. Now
$\Phi$ is surjective because $\Phi(yx)=t$ and $\Phi(y)=p$.
\end{proof}
The next result is \cite[Corollary~3.2]{FerreiraGoncalvesSanchez2}. It will
allow us to obtain free algebras generated by symmetric elements
from free group algebras.
\begin{lem}\label{coro:freeinsidegroupring}
Let $G$ be the free group on the set of two elements $\{x,y\}$. Let
$k$ be a field and consider the group algebra $k[G]$. Then the
$k$-algebra generated by $x+x^{-1}$ and $y+y^{-1}$ inside $k[G]$ is
free on $\{x+x^{-1}, y+y^{-1}\}$. \qed
\end{lem}
Now we are ready to present the main result of this section. It will be used throughout the paper.
Parts (1),(2) and (3) in Sections~\ref{sec:residuallynilpotent} and~\ref{sec:Ore} while
parts (4),(5) in Section~\ref{sec:Heisenberggroup}.
\begin{theo}\label{theo:freegroupHeisenberg}
Let $k$ be a field of characteristic zero. Let $H$ be the
Heisenberg Lie $k$-algebra.
Let $U(H)$ be the universal enveloping algebra of $H$ and
$\mathfrak{D}(H)$ be the Ore division ring of fractions of $U(H)$.
Set $z=[y,x]$, $V=\frac{1}{2}(xy+yx)$, and consider the following
elments of $\mathfrak{D}(H)$:
$$S=(V-\frac{1}{3
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] | 1,024 |
ArXiv
| 0.245968 | 2.35223 | -0.481307 | 0.658748 |
}z)(V+\frac{1}{3}z)^{-1},$$
$$T=(z+y^2)^{-1}(z-y^2)S(z+y^2)(z-y^2)^{-1},$$
$$S_1=z^{-1}\Big((V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}+ (V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)\Big) z^{-1},$$
$$S_2=z\Big((V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}+ (V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)\Big) z,$$
$$T_1=(z+y^2)^{-1}(z-y^2)S_1(z+y^2)(z-y^2)^{-1},$$
$$T_2=(z^2+y^3)^{-1}(z^2-y^3)S_1(z^2+y^3)(z^2-y^3)^{-1},$$
$$T_3=(z+y^2)^{-1}(z-y^2)S_2(z+y^2)(z-y^2)^{-1},$$
$$T_4=(z^2+y^3)^{-1}(z^2-y^3)S_2(z^2+y^3)(z^2-y^3)^{-1}.$$
The following hold true.
\begin{enumerate}[\rm (1)]
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$
is the free group $k$-algebra on the set $\{S,T\}$.
\item
\begin{enumerate}[\rm (a)]
\item The elements $S_1$, $S_1^2$, $T_1$ and $T_1^2$ are symmetric with respect to the involutions in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(2) and (3).
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S_1,T_1\}$ is the free $k$-algebra on the
set $\{S_1,T_1\}$.
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{1+S_1,(1+S_1)^{-1}, 1+T_1,(1+T_1)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1,1+T_1\}$.
\item The $k$ subalgebra of $\mathfrak{D}(H)$ generated by $\{S_1^2,T_1^2\}$ is the free $k$-algebra on the
set $\{S_1^2,T_1^2\}$.
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{1+S_1^2,(1+S_1^2)^{-1}, 1+T_1^2,(1+T_1^2)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1,1+T_1\}$.
\end{enumerate}
\item \begin{enumerate}[\rm (a)]
\item The elements $S_1$, $S_1^2$, $T_2$ and $T_2^2$ are symmetric with respect to the involution in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(1).
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S_1,T_2\}$ is the free $k$-algebra
on the set $\{S_1,T_2\}$.
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{1+S_1,(1+S_1)^{-1}, 1+T_2,(1+T_2)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1,1+T_2\}$.
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S_1^2,T_2^2\}$ is the free $k$-algebra on the
set $\{S
|
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] | 1,024 |
ArXiv
| -0.914896 | 1.874645 | -0.965533 | -0.263979 |
_1^2,T_2^2\}$.
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{1+S_1^2,(1+S_1^2)^{-1}, 1+T_2^2,(1+T_2^2)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1^2,1+T_2^2\}$.
\end{enumerate}
\item
\begin{enumerate}[\rm (a)]
\item The elements $S_2$, $S_2^2$, $T_3$ and $T_3^2$ are symmetric with respect to the involutions in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(2) and (3).
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S_2^2,T_3^2\}$ is the free $k$-algebra on the
set $\{S_2^2,T_3^2\}$.
\end{enumerate}
\item \begin{enumerate}[\rm (a)]
\item The elements $S_2$, $S_2^2$, $T_4$ and $T_4^2$ are symmetric with respect to the involution in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(1).
\item The $k$-subalgebra of $\mathfrak{D}(H)$ generated by $\{S_2^2,T_4^2\}$ is the free $k$-algebra on the
set $\{S_2^2,T_4^2\}$.
\end{enumerate}
\end{enumerate}
\end{theo}
\begin{proof}
Consider the surjective $k$-algebra homomorphism $\Phi\colon
\mathfrak{S}^{-1}U(H)\rightarrow K(p;\sigma)$ given in Lemma~\ref{lem:specializationtoWeyl}.
Then $\Phi(y)=p$, $\Phi(x)=p^{-1}t$ and $\Phi(z)=1$.
Recall that in
$K(p;\sigma)$, we have and $tp=p(t-1)$.
Thus $\Phi(V)=\Phi(\frac{1}{2}(xy+yx))=\frac{1}{2}(p^{-1}tp+pp^{-1}t)=\frac{1}{2}(t-1+t)=t-\frac{1}{2}$.
\begin{eqnarray*}
\Phi(V-\frac{1}{3}z)=t-\frac{1}{2}-\frac{1}{3}=t-\frac{5}{6}, & & \Phi(V+\frac{1}{3}z)=t-\frac{1}{2}+\frac{1}{3}=t-\frac{1}{6}, \\
\Phi(z+y^2)=1+p^2, & & \Phi(z-y^2)=1-p^2,\\
\Phi(z^2+y^3)=1+p^3, & & \Phi(z^2-y^3)=1-p^3.
\end{eqnarray*}
Hence the elements $V-\frac{1}{3}z,\, V+\frac{1}{3}z,\, z+y^2,\, z-y^2,\, z^2+y^3,\, z^2-y^3$ are invertible
in $\mathfrak{S}^{-1}U(H)$.
Thus
$S,S^{-1},T,T^{-1}, S_1,S_1^{-1}, T_1,T_1^{-1}, T_2,T_2^{-1}\in \mathfrak{S}^{-1}U(H)$.
Moreover, following the notation of
Proposition~\ref{prop:freealgebrainWeyl},
$$\Phi(S)=(t-\frac{5}{6})(t-\frac{1}{6})^{-1}=s,$$
$$\Phi(S_1)=(t-\frac{5}{6})(t-\frac{1}{6})^{-1}+(t-\frac{1}{6})(t-\frac{5}{6})^{-1}=s+s^{-1},$$
$$\Phi(S_2)=(t-\frac{5}{6})(t-\frac{1}{6})^{-1}+(t-\frac{1}{6})(t-\frac{5}{6
|
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] | 1,024 |
ArXiv
| -0.544697 | 1.786395 | -0.961947 | -0.154568 |
})^{-1}=s+s^{-1},$$
$$\Phi((z+y^2)^{-1}(z-y^2))=(1+p^2)^{-1}(1-p^2)=(1-p^2)(1+p^2)^{-1}=u,$$
$$\Phi((z+y^2)(z-y^2)^{-1})=(1+p^2)(1-p^2)^{-1}=u^{-1},$$
$$\Phi((z^2+y^3)(z^2-y^3)^{-1})=(1+p^3)(1-p^3)^{-1}={u}_1^{-1}.$$
Hence
$\Phi(T)=usu^{-1}$,
$\Phi(T_1)=u(s+s^{-1})u^{-1}$, $\Phi(T_2)=u_1(s+s^{-1})u_1^{-1}$,
$\Phi(T_3)=u(s+s^{-1})u^{-1}$ and $\Phi(T_4)=u_1(s+s^{-1})u_1^{-1}$.
\medskip
We proceed to show that the elements in statements (1),(2),(3),(4),(5) generate free (group) algebras.
That they are symmetric will be proved below.
\medskip
(1) By Proposition~\ref{prop:freealgebrainWeyl}(1), the set
$\{s,\,s^{-1},\, usu^{-1},\,
us^{-1}u^{-1}\}$ generates a free group $k$-algebra. Therefore, the
$k$-subalgebra of $\mathfrak{S}^{-1}U(H)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$
is the free group $k$-algebra on the
set $\{S,\ T\}$.
(2) By Proposition~\ref{prop:freealgebrainWeyl}(2), the set
$\{s+s^{-1},u(s+s^{-1})u^{-1}\}$ are the free generators of a free $k$-algebra. Therefore the $k$-subagebra generated by $\{S_1,T_1\}$
is the free $k$-algebra on $\{S_1,T_1\}$. This implies that
the $k$-subalgebra generated by $\{S_1^2,T_1^2\}$ is the free $k$-algebra on $\{S_1^2,T_1^2\}$.
Consider $H$ as a $\mathbb{Z}$-graded Lie $k$-algebra as in Example~\ref{ex:gradedLie}(a). Then $U(H)$ is graded according to
Lemma~\ref{lem:gradeduniversalenveloping}. The $k$-algebra $U(H)$ is an Ore domain.
Recall that given a
$\mathbb{Z}$-graded $k$-algebra
which is an Ore domain, localizing at the set of nonzero homogeneous elements yields a graded division ring.
Thus if we localize at the set $\mathcal{H}$ of homogeneous elements of $U(H)$, we get that
$\mathcal{H}^{-1}U(H)$ is a graded division ring.
Notice that $z,\, V-\frac{1}{3}z,\, V+\frac{1}{3}z$ are homogeneous of degree $-2$. Therefore
$S_1$ is homogeneous of degree $4$ and $S_1^2$
is homogeneous of degreee $8$. Notice that $z+y^2,\, z-y^2$ are homogeneous of degree $-2$. Therefore $T_1$ is homogeneous of degree $4$
and $T_1^2$ is homogeneous of degree $8$.
By Theorem~\ref{theo:freegroupgradedOre}, the $k$-subalgebra generated by the set $\{1+S_1,\, (1+S_1)^{-1},\, 1+T_1,\, (1+T_1)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1,1+T_1\}$. Also, by Theorem~\ref{theo:freegroupgradedOre},
the $k$-subalgebra generated by the set $\{1+S_1^2,\, (1+S_1^2)^{-1},\, 1+T_1^2,\, (1+T_1^2
|
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] | 1,024 |
ArXiv
| -0.033889 | 2.040761 | -0.676152 | 0.524671 |
)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1^2,1+T_1^2\}$, as desired.
(3) By Proposition~\ref{prop:freealgebrainWeyl}(3), the set
$\{s+s^{-1},u_1(s+s^{-1})u_1^{-1}\}$ are the free generators of a free $k$-algebra. Therefore the $k$-subagebra generated by $\{S_1,T_2\}$
is the free $k$-algebra on $\{S_1,T_2\}$. This implies that
the $k$-subalgebra generated by $\{S_1^2,T_2^2\}$ is the free $k$-algebra on $\{S_1^2,T_2^2\}$.
Consider $H$ as a $\mathbb{Z}$-graded Lie $k$-algebra as in Example~\ref{ex:gradedLie}(b). Then $U(H)$ is graded according to
Lemma~\ref{lem:gradeduniversalenveloping}. The $k$-algebra $U(H)$ is an Ore domain.
Recall that given a
$\mathbb{Z}$-graded $k$-algebra
which is an Ore domain, localizing at the set of nonzero homogeneous elements yields a graded division ring.
Thus if we localize at the set $\mathcal{H}$ of homogeneous elements of $U(H)$, we get that
$\mathcal{H}^{-1}U(H)$ is a graded division ring.
Notice that $z,\, V-\frac{1}{3}z,\, V+\frac{1}{3}z$ are homogeneous of degree $-3$. Therefore
$S_1$ is homogeneous of degree $6$ and $S_1^2$ is homogeneous
of degree $12$. Notice that $z^2+y^3,\, z^2-y^3$ are homogeneous of degree $-6$.
Therefore $T_2$ is homogeneous of degree $6$ and $T_2^2$ is homogeneous of degree $12$.
By Theorem~\ref{theo:freegroupgradedOre}, the $k$-subalgebra generated by the set $\{1+S_1,\, (1+S_1)^{-1},\, 1+T_2,\, (1+T_2)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1,1+T_2\}$. Also, by
Theorem~\ref{theo:freegroupgradedOre}, the $k$-subalgebra generated by the set $\{1+S_1^2,\, (1+S_1^2)^{-1},\, 1+T_2^2,\, (1+T_2^2)^{-1}\}$
is the free group $k$-algebra on the set $\{1+S_1^2,1+T_2^2\}$, as desired.
(4) By Proposition~\ref{prop:freealgebrainWeyl}(2), the set
$\{s+s^{-1},u(s+s^{-1})u^{-1}\}$ are the free generators of a free $k$-algebra. Therefore the $k$-subagebra generated
by $\{S_2,T_3\}$
is the free $k$-algebra on $\{S_2,T_3\}$. This implies that
the $k$-subalgebra generated by $\{S_2^2,T_3^2\}$ is the free $k$-algebra on $\{S_2^2,T_3^2\}$.
(5) By Proposition~\ref{prop:freealgebrainWeyl}(3), the set
$\{s+s^{-1},u_1(s+s^{-1})u_1^{-1}\}$ are the free generators of a free $k$-algebra.
Therefore the $k$-subagebra generated by $\{S_2,T_4\}$
is the free $k$-algebra on $\{S_2,T_4\}$. This implies that
the $k$-subalgebra generated by $\{S_2^2,T_4^2\}$ is the free $k$-algebra on $\{S_2^2,T_4^2\}$.
\medskip
Now we prove that the elements considered in the statements of (2), (3), (4) and (5) are symmetric.
Consider first the principal involution, that is, the one in
Lemma~\ref
|
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] | 1,024 |
ArXiv
| 0.019773 | 2.279554 | -0.594336 | 0.610683 |
{lem:equivalentinvolutionHeisenbergalgebra}(3).
$$V^* = \frac{1}{2}(xy+yx)^* = \frac{1}{2}(xy+yx) = V$$
$$(V-\frac{1}{3}z)^* = V+\frac{1}{3}z,\qquad (V+\frac{1}{3}z)^* = V-\frac{1}{3}z$$
\begin{eqnarray*}
S_1^*&=& \Big(z^{-1}\Big((V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}+ (V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)\Big) z^{-1}\Big)^*\\
&=& z^{-1}\Big((V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)+(V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1} \Big) z^{-1} \\
&=& S_1 \end{eqnarray*}
\begin{eqnarray*}
((z+y^2)^{-1}(z-y^2))^* & = & (-z-y^2)(-z+y^2)^{-1} \\
&=& (z+y^2)(z-y^2)^{-1}
\end{eqnarray*}
\begin{eqnarray*}
T_1^*&=& ((z+y^2)^{-1}(z-y^2)S(z+y^2)(z-y^2)^{-1})^* \\
&=& (z+y^2)^{-1}(z-y^2)S(z+y^2)(z-y^2)^{-1} \\
&=& T_1.
\end{eqnarray*}
Similarly, $S_2^*=S_2$ and $T_3^*=T_3$.
Consider now the involution in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(2).
$$V^* = \frac{1}{2}(xy+yx)^* =\frac{1}{2}(xy+yx) =V$$
$$(V-\frac{1}{3}z)^*= V+\frac{1}{3}z,\qquad (V+\frac{1}{3}z)^* = V-\frac{1}{3}z$$
\begin{eqnarray*}
S_1^*&=& \Big(z^{-1}\Big((V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}+ (V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)\Big) z^{-1}\Big)^*\\
&=& z^{-1}\Big((V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)+(V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1} \Big) z^{-1} \\
&=& S_1
\end{eqnarray*}
\begin{eqnarray*}((z+y^2)^{-1}(z-y^2))^* & = & (-z-y^2)(-z+y^2)^{-1} \\
&=& (z+y^2)(z-y^2)^{-1}
\end{eqnarray*}
\begin{eqnarray*}T_1^*&=& ((z+y^2)^{-1}(z-y^2)S(z+y^2)(z-y^2)^{-1})^* \\
&=& (z+y^2)^{-1}(z-y^2)S(z+y^2)(z-y^2)^{-1} \\
&=& T_1.\end{eqnarray*}
Similarly $S_2^*=S_2$ and $T_3^*=T_3$.
Finally consider the involution in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(1).
$$V^* = \frac{1}{2}(xy+yx)^* = -\frac{1}{2}(xy+yx) = -V$$
$$(V-\frac{1}{3}z)^* = -V-\frac{1}{3}z,\qquad (V+\frac{1}{3}z)^* = -V+\frac{1}{3}z$$
\begin{eqnarray*}
S_1^*&=& \Big
|
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320,
6970
] | 1,024 |
ArXiv
| -0.483563 | 1.708528 | -1.184306 | -0.149824 |
(z^{-1}\Big((V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}+ (V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)\Big) z^{-1}\Big)^*\\
&=& z^{-1}\Big((-V+\frac{1}{3}z)^{-1}(-V-\frac{1}{3}z)+(-V+\frac{1}{3}z)(-V-\frac{1}{3}z)^{-1} \Big) z^{-1} \\
&=& z^{-1}\Big((V-\frac{1}{3}z)^{-1}(V+\frac{1}{3}z)+ (V-\frac{1}{3}z)(V+\frac{1}{3}z)^{-1}\Big) z^{-1}\\
&=& S_1
\end{eqnarray*}
$$((z^2+y^3)^{-1}(z^2-y^3))^* = (z^2+y^3)(z^2-y^3)^{-1}$$
\begin{eqnarray*}
{T_2}^*&=& ((z^2+y^3)^{-1}(z^2-y^3)S_1(z^2+y^3)(z^2-y^3)^{-1})^* \\
&=& (z^2+y^3)^{-1}(z^2-y^3)S_1(z^2+y^3)(z^2-y^3)^{-1} \\
&=& T_2.
\end{eqnarray*}
Similarly $S_2^*=S_2$ and $T_4^*=T_4$.
\end{proof}
\begin{theo}\label{theo:symmetricnilpotentLie}
Let $k$ be a field of characteristic zero and $L$ be a nonabelian
nilpotent Lie $k$-algebra.
Let $U(L)$ be the universal enveloping algebra of $L$ and
$\mathfrak{D}(L)$ be the Ore ring of fractions of $U(H)$.
Then, for any involution $*\colon L\rightarrow L,$ $f\mapsto f^*$,
there exist nonzero symmetric elements $U,V\in L$ such that the
$k$-subalgebra of $\mathfrak{D}(L)$ generated by $\{U,U^{-1},V,V^{-1}\}$
is the free group $k$-algebra on $\{U,V\}$.
\end{theo}
\begin{proof}
By Proposition~\ref{prop:involutionnilpotent}, there exists a $*$-invariant
Heisenberg Lie $k$-subalgebra $H$ generated by two elements $x,y$ and such that $*$,
when restricted to $H$ is one of the three involutions in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}. Since $U(L)$ is an Ore domain,
then $U(H)$ is also an Ore domain. Thus
the division ring generated by $U(H)$ inside $\mathfrak{D}(L)$ is $\mathfrak{D}(H)$.
By Theorem~\ref{theo:freegroupHeisenberg}, there exist elements $U$ and $V$ as desired.
\end{proof}
\section{Free group algebras in division rings generated by
universal enveloping algebras of residually nilpotent Lie algebras}
\label{sec:residuallynilpotent}
Let $k$ be a field, $L$ be a Lie $k$-algebra and $U(L)$ its universal enveloping
algebra.
It was proved in \cite{Cohnembeddingrings} that $U(L)$ can be embedded in a division ring.
Two similar proofs of this fact were given in \cite{Lichtmanvaluationmethods} and
\cite{Huishiskewfields}. Moreover, the division ring constructed in the foregoing three papers is the same
\cite[Theorem~8]{Huishiskewfields}.
We will work with the construction of the skew field $\mathfrak{D}(L)$ which contains $U(L)$
and it is generated by $U(L)$ (as a division ring) given by Lichtman in~\cite{Lichtmanvaluationmethods}.
The interested reader
|
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ArXiv
| -0.298123 | 1.838307 | -1.195066 | -0.170379 |
can also find this construction in \cite[Section~2.6]{Cohnskew}.
Of course, when $U(L)$ is an Ore domain, $\mathfrak{D}(L)$ is the Ore ring of fractions of $U(L)$.
In this subsection, we want to obtain free
group algebras in $\mathfrak{D}(L)$, where $L$ is a (generalization
of) a residually nilpotent Lie algebra, from the ones obtained in
$\mathfrak{D}(H)$, where $H$ is the Heisenberg Lie algebra. The technique we will use is
from~\cite[Section~4]{FerreiraGoncalvesSanchez2}.
For that, we will need some results on the division ring $\mathfrak{D}(L)$.
For example, $\mathfrak{D}(L)$ is well behaved for Lie subalgebras of $L$ as shown in
\cite[Proposition~2.5]{Lichtmanuniversalfields}. More precisely, if $N$ is a Lie subalgebra
of $L$, then the natural embedding $U(N)\hookrightarrow U(L)$ can be extended to an embedding
$\mathfrak{D}(N)\hookrightarrow \mathfrak{D}(L)$. Furthermore,
if $\mathcal{B}_N$ is a basis of $N$ and $\mathcal{C}$
is a set of elements of $L\setminus N$ such that $\mathcal{B}_N\cup
\mathcal{C}$ is a basis of $L$, then the standard monomials in
$\mathcal{C}$
are linearly independent over
$\mathfrak{D}(N)$. Notice that if $U(L)$ is an Ore domain, then these assertions
are easily verified.
\medskip
Let $k$ be a field and $R$ be a $k$-algebra. Suppose that $\delta\colon R\rightarrow R$
is a $k$-derivation of $R$, that is, $\delta$ is
a $k$-linear map such that $\delta(ab)=\delta(a)b+a\delta(b)$ for all $a,b\in R$.
We will consider the skew polynomial ring
$R[x;\delta]$. The elements of $R[x;\delta]$ are ``right
polynomials'' of the form $\sum_{i=0}^n x^ia_i$, where the
coeficients $a_i$ are in $R$. The multiplication is determined by
$$ax=xa+\delta(a) \quad \textrm{for all } a\in R.$$
Given $R[x;\delta]$, one can construct the \emph{formal
pseudo-differential operator ring}, denoted $R((t_x;\delta))$,
consisting of the formal Laurent series $\sum_{i=n}^\infty
t_x^ia_i$, with $n\in\mathbb{Z}$ and coefficients $a_i\in R$,
satisfying $at_x^{-1}=t_x^{-1}a+\delta(a)$ for all $a\in R$.
Therefore
\begin{equation}
at_x=t_xa-t_x\delta(a)t_x = \sum_{i=1}^\infty
t_x^i(-1)^{i-1}\delta^{i-1}(a),
\end{equation}
for any $a\in R$.
The subset $R[[t_x;\delta]]$ of $R((t_x;\delta))$ consisting of the
Laurent series of the form $\sum_{i=0}^\infty t_x^ia_i$ is a
$k$-subalgebra of $R((t_x;\delta))$.
The set $\mathcal{S}=\{1,t_x,t_x^2,\dots\}$ is a
left denominator set of $R[[t_x;\delta]]$ such that the Ore
localization $\mathcal{S}^{-1}R[[t_x;\delta]]$ is the $k$-algebra
$R((t_x;\delta))$, see for example \cite[Theorem~2.3.1]{Cohnskew}.
If $R$ is a domain, then a series $f\in R((t_x;\delta))$ is invertible
if, and only if, the coefficient of the least element in the support of $f$
is invertible in $R$.
Notice that there is a natural embedding $R[x;\delta]\hookrightarrow
|
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] | 1,024 |
ArXiv
| 0.449069 | 2.456052 | -0.178756 | 0.931961 |
R((t_x;\delta_x))$ sending $x$ to $t_x^{-1}$.
In what follows, $R[y;\delta_y][x;\delta_x]$ means polynomials of the form
$\sum_{i=0}^n x^i f_i$ where each $f_i\in R[y;\delta_y]$ and $\delta_x$ is a $k$-derivation of $R[y;\delta_y]$.
Also $R((t_y;\delta_y))((t_x;\delta_x))$ is the ring of series of the form
$\sum_{i=n}^\infty
t_x^if_i$, with $n\in\mathbb{Z}$, coefficients $f_i\in R((t_y;\delta_y))$ and
$\delta_x$ is a $k$-derivation of $R((y;\delta_y))$.
\medskip
Let $k$ be a field.
Let $L$ a Lie $k$-algebra generated by two elements $u,v$. Let
$H=\langle x,y\mid [[y,x],x]=[[y,x],y]=0\rangle$ be the Heisenberg
Lie $k$-algebra. Suppose that there exists a Lie $k$-algebra
homomorphism \begin{equation*} L\stackrel{\rho}{\rightarrow} H,\
u\mapsto x,\ v\mapsto y.
\end{equation*}
Define $w=[v,u]$ and $z=[y,x]$. Let $N=\ker \rho$. Thus $N$ is a
(Lie) ideal of $L$.
By the universal property of universal enveloping algebras, $\rho$
can be uniquely extended to a $k$-algebra homomorphism $\psi\colon
U(L)\rightarrow U(H)$ between the corresponding universal enveloping
algebras. Note that $\ker \psi$ is the ideal of $U(L)$ generated by
$N$. The restriction $\psi_{|U(N)}$ coincides with the augmentation
map $\varepsilon\colon U(N)\rightarrow k$.
By the PBW-Theorem, the elements of $U(H)$ are uniquely expressed as
finite sums $\sum_{l,m,n\geq 0} x^ly^mz^na_{lmn}$, with
$a_{lmn}\in k$. Let $\delta_x$ be the inner $k$-derivation of $U(H)$
determined by $x$, i.e. $\delta_x(f)=[f,x]=fx-xf$ for all $f\in
U(H)$. It can be proved that
\begin{equation}\label{eq:U(L)aspolynomials}
U(H)=k[z][y][x;\delta_x].
\end{equation}
\begin{equation}\label{eq:U(H)insideseries}
U(H)\hookrightarrow k((t_z))((t_y))((t_x;\delta_x)), \ z\mapsto t_z^{-1},\ y\mapsto t_y^{-1},\
x\mapsto t_x^{-1}.
\end{equation}
Consider now $U(L)$, the universal enveloping algebra of $L$. By
the PBW-Theorem, the elements of $U(L)$ can be uniquely expressed as
finite sums \linebreak $\sum\limits_{l,m,n\geq 0} u^lv^mw^nf_{lmn}$ with $f_{lmn}\in
U(N)$. Since $N$ is an ideal of $L$, the inner derivations
$\delta_u$, $\delta_v$, $\delta_w$ of $U(L)$ defined by $u,v,w$,
respectively, are such that $\delta_u(U(N))\subseteq U(N)$,
$\delta_v(U(N))\subseteq U(N)$, $\delta_w(U(N))\subseteq U(N)$. The
$k$-subalgebra of $U(L)$ generated by $U(N)$ and $w$ is
$U(N)[w;\delta_w]$. Since
$\delta_v(w)\in U(N)\subseteq U(N)[w;\delta_w]$,
the $k$-subalgebra of $U(L)$ generated by $U(
|
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] | 1,024 |
ArXiv
| 0.124889 | 2.26398 | -0.822972 | 0.623332 |
N)$ and
$\{w,v\}$ is $U(N)[w;\delta_w][v;\delta_v]$. Furthermore, since
$\delta_u(v)=w$ and $\delta_u(w)\in U(N)$,
\begin{equation}\label{eq:U(H)aspolynomials}
U(L)=U(N)[w;\delta_w][v;\delta_v][u;\delta_u].
\end{equation}
\begin{eqnarray}\label{eq:U(L)insideseries}
U(L)\hookrightarrow
U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u)), & w\mapsto t_w^{-1},\ v\mapsto t_v^{-1},\ u\mapsto t_u^{-1}, \\
&f\mapsto \varepsilon(f), \textrm{ for all } f\in U(N). \nonumber
\end{eqnarray}
In this setting, the next two lemmas are \cite[Lemmas~4.1,4.2]{FerreiraGoncalvesSanchez2}.
\begin{lem}\label{lem:commutativediagram}
There exists a commutative diagram of embeddings of $k$-algebras
\begin{equation*}\label{eq:diagramnotOre}
\xymatrixcolsep{0.0001cm}\xymatrix{U(L)=U(N)[w;\delta_w][v;\delta_v][u;\delta_u]\ar@{^{(}->}[rr]\ar@{^{(}->}[dd]
\ar@{^{(}->}[rd] & &
\mathfrak{D}(N)[w;\delta_w][v;\delta_v][u;\delta_u]\ar@{^{(}->}[dd]
\ar@{_{(}->}[ld]\\ & \mathfrak{D}(L) \ar@{^{(}->}[dr] &\\
U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))\ar@{^{(}->}[rr]
& & \mathfrak{D}(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))
& }
\end{equation*}\qed
\end{lem}
\begin{lem} \label{lem:morphismsofseries}
Let $\varepsilon\colon U(N)\rightarrow k$ denote the augmentation
map. The following hold true.
\begin{enumerate}[\normalfont (1)]
\item There exists a $k$-algebra homomorphism
$$\Phi_w\colon U(N)((t_w;\delta_w))\rightarrow k((t_z)),\quad \sum_{i}t_w^if_i\mapsto \sum_{i}t_z^i\varepsilon(f_i),$$
where $f_i\in U(N)$ for each $i$.
\item There exists a $k$-algebra homomorphism
$$\Phi_v\colon U(N)((t_w;\delta_w))((t_v;\delta_v))
\rightarrow k((t_z))((t_y)),\quad \sum_{i}t_v^ig_i\mapsto
\sum_{i}t_y^i\Phi_w(g_i),$$ where $g_i\in U(N)((t_w;\delta_w))$ for
each $i$.
\item There exists a $k$-algebra homomorphism
\begin{eqnarray*}
\Phi_u\colon U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))
&\rightarrow& k((t_z))((t_y))((t_x;\delta_x)), \\
\sum_{i}t_u^ih_i &\mapsto& \sum_{i}t_x^i\Phi_v(h_i), \end{eqnarray*}
where $h_i\in U(N)((t_w;\delta_w))((t_y;\delta_y))$ for each $i$ and
extending the embeddings of \eqref{eq:U(H)insideseries} and \eqref{eq:U(L)insideseries}. \qed
\end{enumerate}
|
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] | 1,024 |
ArXiv
| -0.565075 | 1.630661 | -1.383354 | 0.105997 |
\end{lem}
Now we turn our attention to $k$-involutions of $U(L)$ induced from the ones in $L$.
\begin{lem}\label{lem:involutionbehaveswell}
Let $k$ be a field and $L$ be a Lie $k$-algebra generated by two elements $u,v$. Let
$H=\langle x,y\mid [[y,x],x]=[[y,x],y]=0\rangle$ be the Heisenberg
Lie $k$-algebra. Suppose that there exists a Lie $k$-algebra
homomorphism $L\stackrel{\rho}{\rightarrow} H,\
u\mapsto x,\ v\mapsto y.$
Let $N=\ker \rho$.
Consider the induced $k$-algebra homomorphism $\psi\colon U(L)\rightarrow U(H)$.
Suppose that $*\colon L\rightarrow L$ is an involution in $L$ such that
$N$ is a $*$-invariant ideal of $L$ and call again $*$ the induced involution on $H\cong L/N$.
Then $\psi(f^*)=\psi(f)^*$ for all $f\in U(L)$.
\end{lem}
\begin{proof}
Define $w=[v,u]$ and $z=[y,x]$.
Since $*$ is the induced involution on $H\cong L/N$, then $\psi(u^*)=\psi(u)^*=x^*$,
$\psi(v^*)=\psi(v)^*=y^*$, $\psi(w^*)=\psi(w)^*=z^*$ and
$\psi(f^*)=\psi(f)^*=\varepsilon(f)\in k$ for all $f\in U(N)$.
Given $\sum\limits_{l,m,n\geq 0} u^lv^mw^nf_{lmn}$ with $f_{lmn}\in
U(N)$, we have
\begin{eqnarray*}\psi\Big(\sum\limits_{l,m,n\geq 0} u^lv^mw^nf_{lmn}\Big)^* & =&
\Big(\sum\limits_{l,m,n\geq 0} x^ly^mz^n\varepsilon(f_{lmn})\Big)^* \\ & =&
\sum\limits_{l,m,n\geq 0} (z^*)^n(y^*)^m(x^*)^l\varepsilon(f_{lmn})
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\psi\Big(\Big(\sum\limits_{l,m,n\geq 0} u^lv^mw^n f_{lmn}\Big)^*\Big) & = &
\psi\Big(\sum\limits_{l,m,n\geq 0} f_{lmn}^*(w^*)^n(v^*)^m
(u^*)^l\Big) \\ & = &\sum\limits_{l,m,n\geq 0} (z^*)^n(y^*)^m(x^*)^l\varepsilon(f_{lmn}),
\end{eqnarray*}
as desired.
\end{proof}
It is well known that any $k$-involution on $L$ can be extended to
a $k$-involution of $U(L)$. Moreover,
it was proved in \cite[Proposition~5]{Cimpricfreefieldmany} that any $k$-involution of $L$
can be uniquely extended to a $k$-involution of $\mathfrak{D}(L)$. See also
\cite[Proposition~2.1]{FerreiraGoncalvesSanchez2}.
With this in mind, we are ready to prove the main result of this section.
\begin{theo}\label{theo:freesymmetricresiduallynilpotent}
Let $k$ be a field of characteristic zero, let $H=\langle x,y\mid
[[y,x],x]=[[y,x],y]=0\rangle$ be the Heisenberg Lie $k$-algebra and
let $L$ be a Lie $k$-algebra generated by two elements $u,v$.
Suppose that there exists a Lie $k$-algebra homomorphism
\begin{equation*} L{\rightarrow} H,\ u\mapsto x,\ v\mapsto y,
\end{equation*}
|
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] | 1,024 |
ArXiv
| -0.330728 | 1.682573 | -1.22555 | 0.033899 |
with kernel $N$.
Let $w=[v,u]$, $V=\frac{1}{2}(uv+vu)$, and consider the following
elements of $\mathfrak{D}(L)$:
$$S=(V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1},\qquad
T=(w+v^2)^{-1}(w-v^2)S(w+v^2)(w-v^2)^{-1},$$
$$S_1=w^{-1}\Big((V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1}+ (V-\frac{1}{3}w)^{-1}(V+\frac{1}{3}w)\Big) w^{-1},$$
$$T_1=(w+v^2)^{-1}(w-v^2)S_1(w+v^2)(w-v^2)^{-1},$$
$$T_2=(w^2+v^3)^{-1}(w^2-v^3)S_1(w^2+v^3)(w^2-v^3)^{-1}.$$
Then the following hold true.
\begin{enumerate}[\rm(1)]
\item The $k$-subalgebra of
$\mathfrak{D}(L)$
generated by $\{S,S^{-1},T,T^{-1}\}$ is the free group $k$-algebra on the set
$\{S,T\}$.
\item Suppose that $*\colon L\rightarrow L$ is an involution in $L$ such that
$N$ is a $*$-invariant ideal of $L$ and that the induced involution on $H\cong L/N$
is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(2) and (3).
Then the following hold true.
\begin{enumerate}[\rm(a)]
\item The elements $S_1S_1^*$ and $T_1T_1^*$ are symmetric.
\item The $k$-subalgebra of $\mathfrak{D}(L)$
generated by $\{S_1S_1^*, T_1T_1^*\}$ is the free
$k$-algebra on $\{S_1S_1^*, T_1T_1^*\}$.
\item The $k$-subalgebra of $\mathfrak{D}(L)$ generated by
$$\{1+S_1S_1^*, (1+S_1S_1^*)^{-1}, 1+T_1T_1^*,(1+T_1T_1^*)^{-1}\}$$
is the free group $k$-algebra on the set $\{1+S_1S_1^*, 1+T_1T_1^*\}$.
\end{enumerate}
\item Suppose that $*\colon L\rightarrow L$ is an involution in $L$ such that
$N$ is a $*$-invariant ideal of $L$ and that the induced involution on $H\cong L/N$
is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(1).
Then the following hold true.
\begin{enumerate}[\rm(a)]
\item The elements $S_1S_1^*$ and $T_2T_2^*$ are symmetric.
\item The $k$-subalgebra of $\mathfrak{D}(L)$
generated by $\{S_1S_1^*, T_2T_2^*\}$ is the free
$k$-algebra on $\{S_1S_1^*, T_2T_2^*\}$.
\item The $k$-subalgebra of $\mathfrak{D}(L)$ generated by
$$\{1+S_1S_1^*, (1+S_1S_1^*)^{-1}, 1+T_2T_2^*,(1+T_2T_2^*)^{-1}\}$$
is the free group $k$-algebra on the set $\{1+S_1S_1^*, 1+T_2T_2^*\}$.
\end{enumerate}
\end{enumerate}
\end{theo}
\begin{proof}
Define $z=[y,x]\in H$. Consider the embedding $U(H)\hookrightarrow
k((t_z))((t_y))((t_x;\delta_x))$ given in
\eqref{eq:U(H)insideseries}. Since $k((t_
|
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ArXiv
| -0.501903 | 2.056335 | -0.949394 | 0.040223 |
z))((t_y))((t_x;\delta_x))$
is a division $k$-algebra and $U(H)$ is an Ore domain, it extends to
an embedding $\mathfrak{D}(H)\hookrightarrow
k((t_z))((t_y))((t_x;\delta_x))$.
Consider the embedding
$U(L)\hookrightarrow
U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$ given in
\eqref{eq:U(L)insideseries}. Let $\Phi_u\colon
U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u)) \rightarrow
k((t_z))((t_y))((t_x;\delta_x))$ be the homomorphism given in
Lemma~\ref{lem:morphismsofseries}.
Define the following elements in $\mathfrak{D}(H)$:
$V_H=\frac{1}{2}(xy+yx)$,
$$S_H=(V_H-\frac{1}{3}z)(V_H+\frac{1}{3}z)^{-1},$$
$$T_H=(z+y^2)^{-1}(z-y^2)S_H(z+y^2)(z-y^2)^{-1}.$$
$${S_1}_H=z^{-1}\Big((V_H-\frac{1}{3}z)(V_H+\frac{1}{3}z)^{-1}+ (V_H-\frac{1}{3}z)^{-1}(V_H+\frac{1}{3}z)\Big) z^{-1},$$
$${T_1}_H=(z+y^2)^{-1}(z-y^2){S_1}_H(z+y^2)(z-y^2)^{-1},$$
$${T_2}_H=(z^2+y^3)^{-1}(z^2-y^3){S_1}_H(z^2+y^3)(z^2-y^3)^{-1},$$
\medskip
\underline{Claim~1}: The elements $V-\frac{1}{3}w$, $V+\frac{1}{3}w$,
$w+v^2$, $w-v^2$, $w^2+v^3$ and $w^2-v^3$ are all invertible in
$U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
We proceed to prove claim~1. We begin with the element
$w+v^2=t_w^{-1}+t_v^{-2}$. As a series in $t_v$, this element is
invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))$ if and only if
the coefficient of $t_v^{-2}$ is invertible in the ring of
coefficients $U(N)((t_w;\delta_w))$. The coefficient is $1$, which
is clearly invertible. Similarly, it can be proved that $w-v^2$, $w^2+v^3$, and $w^2-v^3$
are invertible. Now we show that $V+\frac{1}{3}w$ is invertible in
$U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$. First we
obtain an expression of $V+\frac{1}{3}w$ as a series in $t_u$.
\begin{eqnarray}
V+\frac{1}{3}w & = & \frac{1}{2} (uv+vu) +\frac{1}{3}w \nonumber\\
& = & \frac{1}{2}(uv+[v,u] +uv) + \frac{1}{3}w \nonumber \\
& = & \frac{1}{2}w+ uv + \frac{1}{3}w \nonumber \\
& = & \frac{5}{6}w +uv \nonumber\\
& = & \frac{5}{6}t_w^{-1} +t_u^{-1}t_v^{-1} \label{eq:coef
|
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] | 1,024 |
ArXiv
| -0.25897 | 1.931747 | -0.999604 | 0.304583 |
series0}.
\end{eqnarray}
Thus, as a series in $t_u$, the coefficient of the least element in
the support of $V+\frac{1}{3}w^3$ is $t_v^{-1}$, which is invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))$.
Hence $V+\frac{1}{3}w^3$ is invertible in
$U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
The case of
$V-\frac{1}{3}w$ is shown analogously, and the claim is proved.
\medskip
(1) By Claim~1, $S$ and $T$ are invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$
and we have $\Phi_u(V)=V_H$, $\Phi_u(S)=S_H$ and
$\Phi_u(T)=T_H$. By Theorem~\ref{theo:freegroupHeisenberg}, the $k$-algebra
generated by $\{S_H,\, S_H^{-1},\, T_H,\, T_H^{-1}\}$ is the free group $k$-algebra on the set
$\{S_H,T_H\}$. By Lemma~\ref{lem:commutativediagram},
$V$, $S$ and $T$ belong to $\mathfrak{D}(L)$. Therefore, the
elements $S$ and $T$ are nonzero and invertible in
$\mathfrak{D}(L)$, and the $k$-subalgebra generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$ is the free group
$k$-algebra on the set $\{S,T\}$.
(2) (a) It is clear.
(b) We will prove in detail the result for the involution in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(2), the
other case can be shown similarly. Let $n_u,n_v,n_w,\in N$ be such that $$u^*=u+n_u,\ v^*=v+n_v,\ w^*=-w+n_w$$
\underline{Claim~2:} The elements $$(V+\frac{1}{3}w)^*,\ (V-\frac{1}{3}w)^*,\ (w+v^2)^*,\ (w-v^2)^*$$
belong to and are invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
From Claim~2, it follows that the elements $S_1^*, T_1^*\in U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
By Lemma~\ref{lem:involutionbehaveswell}, $\Phi_u(Z^*)=\Phi_u(Z)^*$ where $Z$ is any of the elements in Claim~2.
Thus, by Theorem~\ref{theo:freegroupHeisenberg}(2)(a),
\begin{equation}\label{eq:imagephiu}
\Phi_u(S_1^*)=\Phi_u(S_1)^*=S_{1H}^*=S_{1H},\quad \Phi_u(T_1^*)=\Phi_u(T_1)^*=T_{1H}^*=T_{1H}.
\end{equation}
Hence $\Phi_u(S_1S_1^*)=S_{1H}^2$ and $\Phi_u(T_1T_1^*)=T_{1H}^*$.
By Theorem~\ref{theo:freegroupHeisenberg}(2)(d), the $k$-algebra
generated by $\{S_{1H}^2, T_{1H}^2\}$ is the free algebra on $\{S_{1H}^2,T_{1H}^2\}$. Therefore the result follows.
We proceed to prove Claim~2.
\begin{align}\label{eq:coefseries1}
(V+\frac{1}{3}w)^* & = \left(\frac{1}{2
|
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] | 1,024 |
ArXiv
| -0.197592 | 1.978468 | -1.083886 | 0.252723 |
}(uv+vu)+\frac{1}{3}w \right)^* \nonumber \\
& = \frac{1}{2}((u+n_u)(v+n_v)+(v+n_v)(u+n_u)) +\frac{1}{3}(-w+n_w) \nonumber \\
& = \frac{1}{2}(uv+vu+un_v+n_vu+n_uv+vn_u+n_un_v+n_vn_u) \nonumber \\
& \qquad \quad -\frac{1}{3}w+ \frac{1}{3}n_w \nonumber \\
&=\frac{1}{2}(uv+uv+[v,u] +un_v+un_v+[n_v,u]+vn_u+vn_u \nonumber\\
&\qquad\quad+[n_u,v]+n_un_v+n_vn_u)-\frac{1}{3}w+ \frac{1}{3}n_w \nonumber \\
& = u(v+n_v)+ vn_u+\frac{1}{6}w+f_1 \nonumber\\
& = t_u^{-1}(t_v^{-1}+n_v)+t_v^{-1}n_u+\frac{1}{6}t_w^{-1}+f_1,
\end{align}
where $f_1\in U(N)$.
\begin{align}
(V-\frac{1}{3}w)^* & = \left(\frac{1}{2}(uv+vu)-\frac{1}{3}w \right)^* \nonumber\\
& = \frac{1}{2}((u+n_u)(v+n_v)+(v+n_v)(u+n_u)) -\frac{1}{3}(-w+n_w) \nonumber\\
& = \frac{1}{2}(uv+vu+un_v+n_vu+n_uv+vn_u+n_un_v+n_vn_u) \nonumber\\
& \qquad \quad +\frac{1}{3}w- \frac{1}{3}n_w \nonumber\\
&=\frac{1}{2}(uv+uv+[v,u] +un_v+un_v+[n_v,u]+vn_u+vn_u\nonumber\\
&\qquad \quad +[n_u,v]+n_un_v+n_vn_u)+\frac{1}{3}w- \frac{1}{3}n_w \nonumber\\
& = u(v+n_v)+ vn_u+\frac{5}{6}w+f_2 \nonumber\\
& = t_u^{-1}(t_v^{-1}+n_v)+t_v^{-1}n_u+\frac{5}{6}t_w^{-1}+f_2, \label{eq:coefseries2}
\end{align}
where $f_2\in U(N)$. Note that the element $(t_v^{-1}+n_v)$ is invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))$.
Thus $(V+\frac{1}{3})^*$ and $(V-\frac{1}{3})^*$ are invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
There exist $f_3,f_4\in U(N)$ such that
\begin{eqnarray}
(w+v^2)^*&=& -w+n_w+(v+n_v)^2 \nonumber\\
& = & v^2+vn_v+n_vv+n_v^2-w+n_w \nonumber\\
& = & v^2+2vn_v-w+[n_v,v]+ n_v^2+n_w \nonumber\\
& = & t_v^{-2} +2t_v^{-1}n_v -t_w^{-1} + f_3 \label{eq:coefseries3}
\end{eqnarray}
\begin{eqnarray}
(w-v^2)^*&=& -w+n_w-(v+n_v)^2 \
|
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] | 1,024 |
ArXiv
| -0.242593 | 2.134202 | -1.012157 | -0.009107 |
nonumber\\
& = & -v^2-vn_v-n_vv-n_v^2-w+n_w \nonumber\\
& = & -v^2-2vn_v-w-[n_v,v]- n_v^2+n_w \nonumber\\
& = & -t_v^{-2} -2t_v^{-1}n_v -t_w^{-1} + f_4 \label{eq:coefseries4}
\end{eqnarray}
The elements $(w+v^2)^*,(w-v^2)^*$ are invertible because the coefficient of
$t_v^{-2}$ is $\pm1$, which is clearly invertible. And the claim is proved.
\bigskip
(c) By Theorem~\ref{theo:freegroupHeisenberg}(2)(d), the $k$-subalgebra
generated by $\{1+S_{1H}^2, (1+S_{1H})^{-1}, 1+T_{1H}^2, (1+T_{1H})^{-1}\}$ is the free group $k$-algebra
on the set $\{1+S_{1H}^2, 1+T_{1H}^2\}$. Moreover, by \eqref{eq:imagephiu}, $\Phi_u(1+S_1S_1^*)=1+S_{1H}^2$ and
$\Phi_u(1+T_1T_1^*)=1+T_{1H}^2$. Therefore it is enough to prove that the elements
$1+S_1S_1^*$ and $1+T_1T_1^*$ are invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
By \eqref{eq:coefseries0}, $V-\frac{1}{3}w$ and $V+\frac{1}{3}w$ are series of the form
$t_u^{-1}t_v^{-1}(1+h_1)$ where $h_1$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Hence
$(V-\frac{1}{3}w)^{-1}$ and $(V+\frac{1}{3}w)^{-1}$ are series of the form
$t_vt_u(1+h_2)$ where $h_2$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Using that $w^{-1}=t_w$,
we obtain that $S_1$ is a series of the form $2t_w^2+h_3$ where $h_3$
is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$.
By \eqref{eq:coefseries1}, \eqref{eq:coefseries2}, $(V-\frac{1}{3}w)^*$ and
$(V+\frac{1}{3}w)^*$ are series of the form $t_u^{-1}(t_v^{-1}+n_v)(1+h_4)$,
where $h_4$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$.
Hence
$((V-\frac{1}{3}w)^*)^{-1}$ and $((V+\frac{1}{3}w)^*)^{-1}$ are series of the form
$(t_v^{-1}+n_v)^{-1}t_u(1+h_5)$ where $h_5$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Using that $(w^*)^{-1}=(-t_w^{-1}+n_w)^{-1}$,
we obtain that $S_1^*$ is a series of the form $2t_w^2+h_6$ where $h_6$
is a series on positive powers of $t_u$
|
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] | 1,024 |
ArXiv
| -0.498846 | 1.926556 | -1.205825 | -0.214491 |
with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. From these considerations
it follows that $1+S_1S_1^*$ is a series of the form $1+4t_w^4+h_7$ where $h_7$
is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Now $1+4t_w^4$ is
invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))$, and $1+S_1S_1^*$ is
therefore invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
Clearly $(w+v^2)$ and $(w-v^2)$ are series of the form $\pm t_v^{-2}(1+g_1)$
where $g_1$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$. Thus $(w+v^2)^{-1}$ and $(w-v^2)^{-1}$ are
series of the form $\pm t_v^{2}(1+g_2)$
where $g_2$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$. Using that $S_1$ is a series of the form $2t_w^2+h_3$, where $h_3$
is as stated above, we obtain that $T_1$ is a series of the form
$2t_w^2+g_3+h_8$ where $g_3$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$ and $h_8$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$.
By \eqref{eq:coefseries3}, \eqref{eq:coefseries4},
$(w+v^2)^*$ and $(w-v^2)^*$ are series of the form $\pm t_v^{-2}(1+g_4)$
where $g_4$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$. Thus $((w+v^2)^*)^{-1}$ and $((w-v^2)^*)^{-1}$ are
series of the form $\pm t_v^{2}(1+g_5)$
where $g_5$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$. Using that $S_1^*$ is a series of the form $2t_w^2+h_6$, where $h_6$
is as stated above, we obtain that $T_1^*$ is a series of the form
$2t_w^2+g_6+h_9$ where $g_6$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$ and $h_9$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Therefore
$1+T_1T_1^*$ is a series of the form $1+4t_w^4+g_7+h_{10}$ where
$g_7$ is a series on positive powers of $t_v$ and coefficients in
$U(N)((t_w;\delta_w))$ and $h_{10}$ is a series on positive powers of $t_u$ with
coefficients in $U(N)((t_w;\delta_w))((t_v;\delta_v))$. Now
$1+T_1T_1^*$ is invertible because the series $1+4t_w^4+g_
|
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] | 1,024 |
ArXiv
| -0.739646 | 1.885027 | -1.433564 | -0.577669 |
7$ is invertible
in $U(N)((t_w;\delta_w))((t_v;\delta_v))$ since $1+4t_w^4$ is invertible in
$U(N)((t_w;\delta_w))$.
\bigskip
\noindent(3) Suppose that the induced involution on $L/N$ is the one on
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(3).
The result follows very much like (2) from the following claim which can be shown as Claim~2.
\underline{Claim~3:} The elements
$$(V+\frac{1}{3}w)^*,\ (V-\frac{1}{3}w)^*,\ (w^2+v^3)^*,\ (w^3-v^3)^*$$
belong to and are invertible in $U(N)((t_w;\delta_w))((t_v;\delta_v))((t_u;\delta_u))$.
\begin{align*}
(w^2+v^3)^*&= (-w+n_w)^2+(-v+n_v)^3 \\
& = w^2 - wn_w-n_ww +n_w^2- v^3+v^2n_v+vn_vv +n_vv^2\\
& \qquad \quad -vn_v^2-n_v^2v-n_vvn_v+n_v^3\\
& = -v^3+3v^2n_v-v(3n_v^2+[n_v,v])-[n_v^2,v]+[n_v,v^2]\\
& \qquad \quad -[n_v,v]n_v+n_v^3+w^2-2wn_w+[n_w,w]+n_w^2\\
& = -t_v^{-3}+3t_v^{-2}n_v- t_v^{-1}(3n_v^2+[n_v,v])+t_w^{-2}-2t_w^{-1} +f_5
\end{align*}
\begin{align*}
(w^2-v^3)^*&= (-w+n_w)^2-(-v+n_v)^3 \\
& = w^2 - wn_w-n_ww +n_w^2-(- v^3+v^2n_v+vn_vv +n_vv^2-vn_v^2\\
& \qquad\qquad\qquad\qquad -n_v^2v-n_vvn_v+n_v^3 )\\
& = t_v^{-3}-3t_v^{-2}n_v+ t_v^{-1}(3n_v^2+[n_v,v])+t_w^{-2}-2t_w^{-1} +f_6
\end{align*}
The elements $(w^2+v^3)^*,(w^2-v^3)^*$ are invertible because the coefficient of $t_v^{-3}$
is $\pm1$, which is clearly invertible.
\end{proof}
\begin{coro}\label{coro:freesymmetricresiduallynilpotent}
Let $k$ be a field of characteristic zero and $K$ be a
residually nilpotent Lie
$k$-algebra.
Let $u,v\in K$ be such that $[v,u]\neq 0$ and denote by $L$ the Lie
$k$-subalgebra of $K$ generated by $\{u,v\}$.
Let $w=[v,u]$, $V=\frac{1}{2}(uv+vu)$, and consider the following
elements of $\mathfrak{D}(L)$:
$$S=(V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1},\qquad
T=(w+v^2)^{-1}(w-v^2)S(w+v^2)(w-v^2)^{-1},$$
$$S_1=w^{-1}\Big((V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1}+ (V-\frac{1}{3}w)^{-
|
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ArXiv
| -0.432618 | 1.708528 | -1.150235 | -0.126108 |
1}(V+\frac{1}{3}w)\Big) w^{-1},$$
$$T_1=(w+v^2)^{-1}(w-v^2)S_1(w+v^2)(w-v^2)^{-1},$$
$$T_2=(w^2+v^3)^{-1}(w^2-v^3)S_1(w^2+v^3)(w^2-v^3)^{-1}.$$
Then the following hold true.
\begin{enumerate}[\rm(1)]
\item The Lie $k$-algebra $L/[[L,L],L]$ is isomorphic to $H$, the Heisenberg Lie
$k$-algebra.
\item The $k$-subalgebra of $\mathfrak{D}(L)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$ is the free group
$k$-algebra on the set $\{S,T\}$.
\item Suppose that $L$ is invariant under $*$ and that the induced involution on
$L/[[L,L],L]$ is one of the involution on Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}.
\begin{enumerate}[\rm(i)]
\item If the induced involution on $L/[[L,L],L]$
is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(2) and (3).
Then the following hold true.
\begin{enumerate}[\rm(a)]
\item The elements $S_1S_1^*$ and $T_1T_1^*$ are symmetric.
\item The $k$-subalgebra of $\mathfrak{D}(L)$
generated by $\{S_1S_1^*, T_1T_1^*\}$ is the free
$k$-algebra on $\{S_1S_1^*, T_1T_1^*\}$.
\item The $k$-subalgebra of $\mathfrak{D}(L)$ generated by
$$\{1+S_1S_1^*, (1+S_1S_1^*)^{-1}, 1+T_1T_1^*,(1+T_1T_1^*)^{-1}\}$$
is the free group $k$-algebra on the set $\{1+S_1S_1^*, 1+T_1T_1^*\}$.
\end{enumerate}
\item If the induced involution on $L/[[L,L],L]$
is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}~(1).
Then the following hold true.
\begin{enumerate}[\rm(a)]
\item The elements $S_1S_1^*$ and $T_2T_2^*$ are symmetric.
\item The $k$-subalgebra of $\mathfrak{D}(L)$
generated by $\{S_1S_1^*, T_2T_2^*\}$ is the free
$k$-algebra on $\{S_1S_1^*, T_2T_2^*\}$.
\item The $k$-subalgebra of $\mathfrak{D}(L)$ generated by
$$\{1+S_1S_1^*, (1+S_1S_1^*)^{-1}, 1+T_2T_2^*,(1+T_2T_2^*)^{-1}\}$$
is the free group $k$-algebra on the set $\{1+S_1S_1^*, 1+T_2T_2^*\}$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{coro}
\begin{proof}
Define $N=[[L,L],L]$.
Since $L$ is residually nilpotent and not abelian, $[v,u]\in [L,L]\setminus N$.
Thus $L/N$ is not abelian.
Moreover
$L/N$ is a noncommutative 3-dimensional Lie $k$-algebra with basis
$\{\bar{u},\bar{v},\bar{w}\}$, the classes of $u$, $v$ and $w$ in
$L/N$. Moreover $[L/N,L/N]=k\bar{w}$ which is contained in the
center of $L/N$. Therefore $L/N$ is the Heisenberg Lie $k$-
|
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ArXiv
| -0.667644 | 1.916174 | -1.021123 | -0.216547 |
algebra.
By Theorem~\ref{theo:freesymmetricresiduallynilpotent}, the result
holds for $\mathfrak{D}(L)$. Since
$\mathfrak{D}(L)\hookrightarrow\mathfrak{D}(K)$, the result
follows.
\end{proof}
\begin{coro}\label{coro:symmetricresiduallynilpotent}
Let $k$ be a field of characteristic zero and $L$ be a nonabelian residually nilpotent
Lie $k$-algebra endowed with
an involution $*\colon L\rightarrow L$.
Then there exist symmetric elements $A,B\in \mathfrak{D}(L)$ such that
the $k$-subalgebra of $\mathfrak{D}(L)$ generated by $\{A,A^{-1},B,B^{-1}\}$ is the free
group $k$-algebra on $\{A,B\}$.
\end{coro}
\begin{proof}
Let $N$ be the $*$-invariant ideal $[L,[L,L]]$.
The Lie $k$-algebra $L/N$ is nilpotent but not abelian and $*$ induces an involution
on $L/N$. By Proposition~\ref{prop:involutionnilpotent}, there exists an invariant
Heisenberg Lie $k$-subalgebra $H$ of $L/N$ such that the restriction of the involution is
one of involutions of Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}. Let
$a+N$, $b+N$ be the generators of $H$. Let $M$ be the Lie $k$-subalgebra of $L$ generated
by $N\cup\{a,b\}$. Then $*$ induces an involution $*\colon M\rightarrow M$ by restriction,
$M/N\cong H$ and the induced involution on $M/N$ is one of the involutions of Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}. Apply Theorem~\ref{theo:freesymmetricresiduallynilpotent}(2) and (3)
to obtain that $\mathfrak{D}(M)$ satisfies the
desiered result. Now observe that $\mathfrak{D}(M)\subseteq \mathfrak{D}(L)$.
\end{proof}
\section{Free group algebras in the Ore ring of fractions of universal enveloping
algebras which are Ore domains}\label{sec:Ore}
The main results in this section are
Theorems~\ref{theo:freegroupOre},\ref{theo:symmetricOre1},\ref{theo:symmetricOre2}. They
all have a similar but technical proof. Thanks to
the results in Section~\ref{sec:freegroupalgebrasdivision},
the method can be seen as an improvement of the technique
originally used in the proof of \cite[Theorem~2]{Lichtmanfreeuniversalenveloping} and that
was also used to show \cite[Theorem~5.2]{FerreiraGoncalvesSanchez2}.
\subsection{On conjecture (GA)}
\begin{theo}\label{theo:freegroupOre}
Let $k$ be a field of characteristic zero and $L$ be a Lie
$k$-algebra whose universal enveloping algebra $U(L)$ is an
Ore domain. Let $u,v\in L$ such that the Lie subalgebra generated by
them is of dimension at least three.
Define $w=[v,u]$, $V=\frac{1}{2}(uv+vu)$, and consider the
following elements of $\mathfrak{D}(L)$ the Ore ring of
fractions of $U(L)$:
$$S=(V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1},\qquad T=(w+v^2)^{-1}(w-v^2)S(w+v^2)(w-v^2)^{-1}.$$ Then the $k$-subalgebra
of $\mathfrak{D}(L)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$ is the free group
$k$-algebra on $\{S,T\}$.
\end{theo}
\begin{proof}
Let
|
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] | 1,024 |
ArXiv
| -0.081438 | 1.744866 | -0.798763 | 0.32735 |
$L_1$ be the Lie $k$-subalgebra of $L$ generated by $u$ and $v$.
Since $U(L)$ is an Ore domain, $U(L_1)$ is also an Ore domain and
$\mathfrak{D}(L_1)\subseteq \mathfrak{D}(L)$. Thus, we may
suppose that $L$ is generated by $u$ and $v$.
Consider the filtration $F_{\mathbb{Z}}L=\{F_nL\}_{n\in\mathbb{Z}}$ of $L$
given in
Example~\ref{ex:usualfiltrationLiealgebra}. It induces a filtration
$F_\mathbb{Z}U(L)=\{F_nU(L)\}_{n\in\mathbb{Z}}$ on $U(L)$ as shown
in Section~\ref{sec:filtrationuniversal}. Moreover, by
Lemma~\ref{lem:filtrationuniversalenveloping}(1), there exists
an isomorphism of $\mathbb{Z}$-graded $k$-algebras
\begin{equation}\label{eq:isomorphismofgraded}
U(\gr_{F_\mathbb{Z}}(L)) \cong \gr_{F_\mathbb{Z}}(U(L)),
\end{equation}
which induces a valuation $\upsilon\colon U(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ as in Section~\ref{sec:generalfiltrations}.
It can be
extended to a valuation $\upsilon\colon \mathfrak{D}(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ \cite[Proposition~9.1.1]{Cohnskew}.
We recall that the
filtration it induces is $F_\mathbb{Z}\mathfrak{D}(L)=\{F_n\mathfrak{D}(L)\}_{n\in\mathbb{Z}}$
where $F_n\mathfrak{D}(L)=\{f\in \mathfrak{D}(L)\colon \upsilon(f)\geq n\}$.
In what follows, the two
objects in \eqref{eq:isomorphismofgraded} will be identified.
Consider $u,v$ and $w=[v,u]$. Note that $\upsilon(u)=\upsilon(v)=-1$
and $\upsilon(w)=-2$ because $L$ is not two-dimensional. Denote by
$\bar{u}$, $\bar{v}$ the class of $u,v\in U(L)_{-1}$ and also the
class of $u$ and $v$ in $L_{-1}$. Denote by $\bar{w}$ the class of
$w$ in $U(L)_{-2}$ and in $L_{-2}$. By
Lemma~\ref{lem:gradedOre}(4), $U(\gr_{F_\mathbb{Z}}(L))$ is an Ore domain.
Let $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$ be its Ore ring of fractions.
Now, $\gr_{F_\mathbb{Z}}(L)$ is a (negatively) graded Lie $k$-algebra
which is not abelian $(w\in L_{-2}\setminus L_{-1})$. Thus
$\gr_{F_\mathbb{Z}}(L)$ is a nonabelian residually nilpotent Lie $k$-algebra.
Observe that $[\bar{v},\bar{u}]=\bar{w}$ as elements of
$\gr_{F_\mathbb{Z}}(L)$.
Now define
$\overline{V}=\frac{1}{2}(\bar{u}\bar{v}+\bar{v}\bar{u})$,
$$\overline{S}=(\overline{V}-\frac{1}{3}\bar{w})(\overline{V}+\frac{1}{3}\bar{w})^{-1},\qquad
\overline{T}=(\bar{w}+\bar{v}^2)^{-1}(\bar{w}-\bar{v}^2)\overline{S}(\bar{w}+\bar{v}^2)(\bar{w}-\bar{v}^2)^{-1}.$$
Then Corollary~\ref{coro:freesymmetricresiduallynilpotent}(2) shows
that the $k$-subalgebra of $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$ generated
by
$\{\overline{S},{\overline{S}^{\phantom{.}}}^{-1},\overline{T},{\overline{T}^{\phantom{.}}}^{-1}\}$
is the free
|
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] | 1,024 |
ArXiv
| 0.209967 | 2.362612 | -0.759312 | 0.331145 |
group $k$-algebra on $\{\overline{S},\overline{T}\}$.
Let $\mathcal{H}$ be the set of homogeneous elements of $\gr_{F_\mathbb{Z}}(U(L))$. From
\eqref{eq:isomorphismofgraded}, and Lemma~\ref{lem:gradedOre} we
obtain the following commutative diagram
$$\xymatrix{\gr_{F_\mathbb{Z}}(U(L))\cong U(\gr_{F_\mathbb{Z}}(L))\ar@{^{(}->}[r]\ar@{^{(}->}[d]
&
\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))\\
\mathcal{H}^{-1}\gr_{F_\mathbb{Z}}(U(L))\cong\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))
\ar@{^{(}->}[ur] & },$$ where the diagonal arrow is obtained from the
universal property of the Ore localization. Note that
$\overline{V},\, \overline{V}-\frac{1}{3}\bar{w},\,
\overline{V}+\frac{1}{3}\bar{w},\, \bar{w}+\bar{v}^2,\,
\bar{w}-\bar{v}^2$ are homogeneous elements of degree $-2$ in
$\gr_{F_\mathbb{Z}}(U(L))$. Thus $\overline{S}$,
${\overline{S}^{\phantom{.}}}^{-1}$, $\overline{T}$,
${\overline{T}^{\phantom{.}}}^{-1}$ are in fact homogeneous elements
of degree zero in $\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$.
Now observe that $S$ and $T$ are elements of $\mathfrak{D}(L)$ such
that $\upsilon(S)=\upsilon(T)=0$ and
$\overline{S}=S+\mathfrak{D}(L)_{>0},\,
\overline{T}=T+\mathfrak{D}(L)_{>0}$ in
$\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$. By
Proposition~\ref{prop:freeobjecthomogeneous}, the $k$-subalgebra of
$\mathfrak{D}(L)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$ is
the free group $k$-algebra on $\{S,T\}$.
\end{proof}
When the Lie subalgebra generated by $u$ and $v$ is of dimension
two, we cannot apply the methods developed thus far, but we have the
following consequence of Cauchon's Theorem.
\begin{prop}\label{prop:twodimensionalcase}
Let $k$ be a field of characteristic zero. Let $M$ be the
nonabelian two dimensional Lie $k$-algebra. Thus $M$ has a basis
$\{e,f\}$ such that $[e,f]=f$. Define
$s=(e-\frac{1}{3})(e+\frac{1}{3})^{-1}$ and $u=(1-f)(1+f)^{-1}$.
Consider the embedding $U(M)\hookrightarrow\mathfrak{D}(M)$. Then
the $k$-algebra generated by the set
$\{S=s,\, S^{-1}, T=usu^{-1},\, T^{-1}\}$ is
the free group $k$-algebra on $\{S,\ T\}$.
\end{prop}
\begin{proof}
Since $[e,f]=ef-fe=f$, $ef=f(e+1)$. Thus $U(M)$ can be seen as a
skew polynomial $k$-algebra, $U(M)=k[e][f;\sigma]$, where
$\sigma(e)=e+1$.
According to Cauchon's Theorem, if we define
$s=(e-\frac{1}{3})(e+\frac{1}{3})^{-1}$ and $u=(1-f)(1+f)^{-1}$, the $k$-subalgebra
generated by $\{s,\, s^{-1},\, usu^{-1},\, us^{-1}u^{-1}\}$ is the free group $k$-algebra
on $\{s,\, usu^{-1}\}$.
\end{proof}
Combining together Theorem~\ref{theo:freegroupOre} and
Proposition~\ref{prop:twodimensionalcase}, we obtain the following
result which is \cite[Theorem~4]{Lichtmanfreeuniversalenveloping}.
\begin{theo}\label{theo:frees
|
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] | 1,024 |
ArXiv
| 0.127776 | 2.300318 | -0.692963 | 0.476606 |
ymmetricOregeneral}
Let $k$ be a field of characteristic zero. Let $L$ be a
noncommutative Lie \mbox{$k$-algebra} such that $U(L)$ is an Ore
domain. Then there exist elements $S,T\in\mathfrak{D}(L)$ such that
the $k$-subalgebra of $\mathfrak{D}(L)$ generated by $\{S,\, S^{-1},\, T,\, T^{-1}\}$ is the
free group $k$-algebra on $\{S,T\}$.
More precisely, let $u,v\in L$ such that $[v,u]\neq 0$. Then
\begin{enumerate}[\normalfont(1)]
\item if the Lie $k$-subalgebra of $L$ generated by $\{u,v\}$ is of dimension greater than two, then
one can choose $S$ and $T$ as defined in
Theorem~\ref{theo:freegroupOre},
\item if the Lie $k$-subalgebra of $L$ generated by $\{u,v\}$ is of dimension exactly two, then
one can choose $S$ and $T$ as defined in
Proposition~\ref{prop:twodimensionalcase}. \qed
\end{enumerate}
\end{theo}
\subsection{On involutional versions of conjecture (GA)}
Now we turn our attention involutions and the existence of free group
algebras generated by symmetric elements.
\begin{theo}\label{theo:symmetricOre1}
Let $k$ be a field of characteristic zero and $L$ be a Lie $k$-algebra such that
$U(L)$ is an Ore domain. Let $*\colon L\rightarrow L$ be a $k$-involution.
Suppose that there exists an element $a\in L$ such that $[a^*,a]\neq 0$
and the Lie $k$-subalgebra generated by $\{a,a^*\}$ is of dimension at least $3$.
Define $u=a+a^*$, $v=a^*-a$, $w=[v,u]$ and $V=\frac{1}{2}(uv+vu)$, and consider the following
elements of $\mathfrak{D}(L)$:
$$S_1=w^{-1}\Big((V-\frac{1}{3}w)(V+\frac{1}{3}w)^{-1}+ (V-\frac{1}{3}w)^{-1}(V+\frac{1}{3}w)\Big) w^{-1},$$
$$T_2=(w^2+v^3)^{-1}(w^2-v^3)S_1(w^2+v^3)(w^2-v^3)^{-1}.$$
Then the $k$-subalgebra of $\mathfrak{D}(L)$ generated by
$$\{1+S_1S_1^*, (1+S_1S_1^*)^{-1}, 1+T_2T_2^*,(1+T_2T_2^*)^{-1}\}$$
is the free group $k$-algebra on the set $\{1+S_1S_1^*, 1+T_2T_2^*\}$.
\end{theo}
\begin{proof}
Let $L_1$ be the Lie $k$-subalgebra of $L$ generated by $u$ and $v$.
Since $U(L)$ is an Ore domain, $U(L_1)$ is also an Ore domain.
Moreover, $\mathfrak{D}(L_1)\subseteq \mathfrak{D}(L)$. Thus, we may
suppose that $L$ is generated by $u$ and $v$.
Consider the filtration $F_{\mathbb{Z}}L=\{F_nL\}_{n\in\mathbb{Z}}$ of $L$
defined by
$F_rL=0$ for all $r\geq 0$, $F_{-1}L=ku$, $F_{-2}L=kv+F_{-1}L$, $F_{-3}L=k[v,u]+F_{-2}L$
and for $n\leq -3$,
$$F_{n-1}L=\sum_{n_1+n_2+\dotsb+n_r\geq (n-1)}[F_{n_1}L,[F_{n_2}L,\dotsc] \dotsb].$$
Observe that, for each $n\in\mathbb{Z}$, there exists $\mathcal{
|
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] | 1,024 |
ArXiv
| -0.322577 | 2.17054 | -0.882148 | -0.016696 |
B}_{n}\subseteq L$
whose classes give a basis of $L_{n}=F_nL/F_{n+1}L$ such that
$\bigcup\limits_{n\in\integers} \mathcal{B}_{n}$ is a basis of $L$.
This filtration on $L$ induces a filtration
$F_\mathbb{Z}U(L)=\{F_nU(L)\}_{n\in\mathbb{Z}}$ on $U(L)$ as shown
in Section~\ref{sec:filtrationuniversal}. Moreover, by
Lemma~\ref{lem:filtrationuniversalenveloping}, there exists an isomorphism of $\mathbb{Z}$-graded
$k$-algebras
\begin{equation}\label{eq:isomorphismofgraded2}
U(\gr_{F_\mathbb{Z}}(L)) \cong \gr_{F_\mathbb{Z}}(U(L)),
\end{equation}
which induces a valuation $\upsilon\colon U(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ as in Section~\ref{sec:generalfiltrations}.
In what follows, the two
objects in \eqref{eq:isomorphismofgraded2} will be identified
via the isomorphism given in either \cite[Proposition~1]{Vergne} or
\cite[Lemma~2.1.2]{Boiscorpsenveloppants}. This isomorphism sends
the class of an element of $\mathcal{B}_n$ in $L_n$ to
its class in $U(L)_n$.
Note that each $F_nL$ is invariant under $*$ because $u^*=u$ and $v^*=-v$.
Hence $*$ induces an involution on $\gr_{F_\mathbb{Z}}(L)$ and hence
on $U(\gr_{F_\mathbb{Z}}(L))$. Moreover each $F_nU(L)$ is invariant under $*$,
and thus $*$ also induces an involution on $\gr_{F_\mathbb{Z}}(U(L))$. Therefore
the isomorphism given in \eqref{eq:isomorphismofgraded2} is an isomorphism of
$k$-algebras with involution, that is, if $\Phi$ is the isomorphism of \eqref{eq:isomorphismofgraded2}
then $\Phi(f^*)=\Phi(f)^*$.
Observe that
$\gr_{F_\mathbb{Z}}(L)$ is a residually nilpotent Lie $k$-algebra. Define
$N=\bigoplus_{n\geq 4}L_n$. Then
$\gr_{F_\mathbb{Z}}(L)/N$ is isomorphic to the Heisenberg Lie $k$-algebra $H$. Moreover $N$
is invariant under the involution $*$, and the induced involution in $\gr_{F_\mathbb{Z}}(L)/N$
is the one in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(1).
The valuation $\upsilon\colon U(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ can be
extended to a valuation $\upsilon\colon \mathfrak{D}(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ \cite[Proposition~9.1.1]{Cohnskew}.
Consider $u,v$ and $w=[v,u]$. Note that $\upsilon(u)=-1,\upsilon(v)=-2$
and $\upsilon(w)=-3$. Denote by
$\bar{u}$, $\bar{v}$, $\bar{w}$ the class of $u\in U(L)_{-1}$, $v\in U(L)_{-2}$,
$w\in U(L)_{-3}$ and also the
class of $u$ in $L_{-1}$, $v\in L_{-2}$ and $w\in L_{-3}$, respectively. By
Lemma~\ref{lem:gradedOre}(4), $U(\gr_{F_\mathbb{Z}}(L))$ is an Ore domain.
Let $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$ be its Ore ring of fractions.
Observe that $[\bar{v},\bar{u}]=\bar{w}$ as elements of
$\gr_{F_\mathbb{Z}}(L)$.
Define
$\overline{V}=\frac{1}{2}(\bar{u}\bar{v}+\bar
|
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] | 1,024 |
ArXiv
| 0.409671 | 2.456052 | -0.811764 | 0.524671 |
{v}\bar{u})$, and consider the following
elements of $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$:
$$\overline{S}_1=\bar{w}^{-1}\Big((\overline{V}-\frac{1}{3}\bar{w})
(\overline{V}+\frac{1}{3}\bar{w})^{-1}+ (\overline{V}-\frac{1}{3}\bar{w})^{-1}(\overline{V}+\frac{1}{3}\bar{w})\Big) \bar{w}^{-1},$$
$$\overline{T}_2=(\bar{w}^2+\bar{v}^3)^{-1}(\bar{w}^2-\bar{v}^3)\overline{S}_1(\bar{w}^2+\bar{v}^3)(\bar{w}^2-\bar{v}^3)^{-1}.$$
By Corollary~\ref{coro:freesymmetricresiduallynilpotent}(3)(ii)(b),
the $k$-subalgebra of $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$
generated by $\{\overline{S}_1\overline{S}_1^*, \overline{T}_2\overline{T}_2^*\}$ is the free
$k$-algebra on $\{\overline{S}_1\overline{S}_1^*, \overline{T}_2\overline{T}_2^*\}$.
Let $\mathcal{H}$ be the set of homogeneous elements of $\gr_{F_\mathbb{Z}}(U(L))$. From
\eqref{eq:isomorphismofgraded2} and Lemma~\ref{lem:gradedOre}, we
obtain the following commutative diagram
$$\xymatrix{\gr_{F_\mathbb{Z}}(U(L))\cong U(\gr_{F_\mathbb{Z}}(L))\ar@{^{(}->}[r]\ar@{^{(}->}[d]
&
\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))\\
\mathcal{H}^{-1}\gr_{F_\mathbb{Z}}(U(L))\cong\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))
\ar@{^{(}->}[ur] & },$$ where the diagonal arrow is obtained from the
universal property of the Ore localization. Note that
$\overline{V},\, \overline{V}-\frac{1}{3}\bar{w},\,
\overline{V}+\frac{1}{3}\bar{w}$ are homogeneous elements of degree $-3$,
and the elements $\bar{w}^2+\bar{v}^3,\,
\bar{w}^2-\bar{v}^3$ are homogeneous elements of degree $-3$ in
$\gr_{F_\mathbb{Z}}(U(L))$. Thus $\overline{S}_1$,
$\overline{S}_1^*$, $\overline{T}_2$,
$\overline{T}_2^*$ are in fact homogeneous elements
of degree $-6$ in $\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$.
Now observe that $S_1$, $S_1^*$, $T_2$ and $T_2^*$ are elements of $\mathfrak{D}(L)$ such
that $\upsilon(S_1)=\upsilon(S_1^*)=\upsilon(T_2)=\upsilon(T_2^*)=6$,
hence $\upsilon(S_1S_1^*)=12$, $\upsilon(T_2T_2^*)=12$ and
$\overline{S}_1\overline{S}_1^*=S_1S_1^*+\mathfrak{D}(L)_{>12},\,
\overline{T}_2\overline{T}_2^*=T_2T_2^*+\mathfrak{D}(L)_{>12}$ in
$\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$.
Now, by Theorem~\ref{coro:divisionrings}, the result follows.
\end{proof}
In case that $[x,x^*]=0$ for all $x\in L$ we are able to prove the following.
\begin{theo}\label{theo:symmetricOre2}
Let $k$ be a field of characteristic zero and $L$ be a Lie $k$-algebra such that
$U(L)$ is an Ore domain. Let $*\colon L\rightarrow L$ be a $k$-involution.
Suppose that $[x,x^*]=
|
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] | 1,024 |
ArXiv
| -0.211857 | 2.248407 | -0.984362 | -0.195044 |
0$ for all $x\in L$, but there exist elements
$x,y\in L$ such that $[y,x]\neq 0$ and the $k$-subspace of $L$ spanned by
$\{x,x^*,y,y^*\}$ is not the Lie $k$-subalgebra generated by $\{x,x^*,y,y^*\}$.
Then there exist symmetric elements $A,B\in\mathfrak{D}(L)$ such that the $k$-subalgebra
generated by $\{A,A^{-1},B,B^{-1}\}$ is the free group $k$-algebra on $\{A,B\}$.
\end{theo}
\begin{proof}
Let $L_1$ be the Lie $k$-subalgebra of $L$ generated by $\{x,x^*,y,y^*\}$.
Since $U(L)$ is an Ore domain, $U(L_1)$ is also an Ore domain.
Moreover, $\mathfrak{D}(L_1)\subseteq \mathfrak{D}(L)$. Thus, we may
suppose that $L$ is generated by $\{x,x^*,y,y^*\}$. Let $V$ be the $k$-subspace
spanned by $\{x,x^*,y,y^*\}$.
Consider the filtration $F_{\mathbb{Z}}L=\{F_nL\}_{n\in\mathbb{Z}}$ of $L$
defined by
$F_rL=0$ for all $r\geq 0$, $F_{-1}L=V$, $F_{-2}L=[V,V]+F_{-1}L$,
and for $n\leq -2$,
$$F_{n-1}L=\sum_{n_1+n_2+\dotsb+n_r\geq (n-1)}[F_{n_1}L,[F_{n_2}L,\dotsc] \dotsb].$$
Note that $F_{n}L$ is invariant under $*$ for all $n\in\mathbb{Z}$. Thus the involution
on $L$ induces an involution on $\gr_{F_\mathbb{Z}}(L)$. Define now
$N=\bigoplus_{n\leq -3}L_n$. Then $N$ is an ideal of $\gr_{F_\mathbb{Z}}(L)$ such that
$\gr_{F_\mathbb{Z}}(L)/N$ is a nonabelian nilpotent Lie $k$-algebra because
$[V,V]$ is not contained in $V$ by assumption. Moreover $N$
is invariant under $*$ and thus the involution on $\gr_{F_\mathbb{Z}}(L)$ induces an involution
on $\gr_{F_\mathbb{Z}}(L)/N$ again denoted by $*$. By Proposition~\ref{prop:involutionnilpotent},
there exist $u,v\in \gr_{F_\mathbb{Z}}(L)/N$ such that they generate
a $*$-invariant Heisenberg Lie $k$-subalgebra of $\gr_{F_\mathbb{Z}}(L)/N$
and the restriction to it is one of the involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}.
Note that $F_{-1}L=L_{-1}$. Also
$\gr_{F_\mathbb{Z}}(L)/N\cong L_{-1}\oplus L_{-2}$ as $k$-vector spaces,
and the induced product $[L_{-1},L_{-2}]=0$.
Thus we can choose $u,v\in L_{-1}=F_{-1}L$.
Suppose that the involution on $\gr_{F_\mathbb{Z}}(L)/N$ is like the one in
Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(1), i.e. $u^*=u$, $v^*=-v$
and $w^*=w$, where $w=[v,u]$. Then take $u_1=u+v,\, v_1=u-v\in L_{-1}$.
Note that $u_1^*=v_1$ and $[u_1,v_1]=2[v,u]\neq 0$, a contradiction
with our assumption that $[x,x^*]=0$ for all $x\in L$. Hence the involution on
the Heisenberg subalgebra of $\gr_{F_\mathbb{Z}}(L)/N$ generated by $u,v$
is of type either Lemma~\ref{lem:equ
|
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ArXiv
| -0.028455 | 2.331465 | -0.997811 | 0.23122 |
ivalentinvolutionHeisenbergalgebra}(2)
or Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(3).
Let $L_2$ be the Lie $k$-subalgebra of $L$ generated by $\{u,v\}$.
Since $U(L)$ is an Ore domain, $U(L_2)$ is also an Ore domain.
Moreover, $\mathfrak{D}(L_2)\subseteq \mathfrak{D}(L)$. Thus, we may
suppose that $L$ is generated by $\{u,v\}$. Let $V$ be the $k$-subspace
spanned by $\{u,v\}$.
Consider the filtration $F_{\mathbb{Z}}L=\{F_nL\}_{n\in\mathbb{Z}}$ of $L$
defined by
$F_rL=0$ for all $r\geq 0$, $F_{-1}L=V$, $F_{-2}L=[V,V]+F_{-1}L$,
and for $n\leq -2$,
$$F_{n-1}L=\sum_{n_1+n_2+\dotsb+n_r\geq (n-1)}[F_{n_1}L,[F_{n_2}L,\dotsc] \dotsb].$$
Note that $F_{n}L$ is invariant under $*$ for all $n\in\mathbb{Z}$. Thus the involution
on $L$ induces an involution on $\gr_{F_\mathbb{Z}}(L)$. Define now
$N=\bigoplus_{n\leq -3}L_n$. Then $N$ is an ideal of $\gr_{F_\mathbb{Z}}(L)$ such that
$\gr_{F_\mathbb{Z}}(L)/N$ is the Heisenberg Lie $k$-algebra and the involution induced on it
is of type either Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(2)
or Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(3).
Observe that, for each $n\in\mathbb{Z}$, there exists $\mathcal{B}_{n}\subseteq L$
whose classes give a basis of $L_{n}=F_nL/F_{n+1}L$ such that
$\bigcup\limits_{n\in\integers} \mathcal{B}_{n}$ is a basis of $L$.
This filtration on $L$ induces a filtration
$F_\mathbb{Z}U(L)=\{F_nU(L)\}_{n\in\mathbb{Z}}$ on $U(L)$ as shown
in Section~\ref{sec:filtrationuniversal}. Moreover, by
Lemma~\ref{lem:filtrationuniversalenveloping}, there exists an isomorphism of $\mathbb{Z}$-graded $k$-algebras
\begin{equation}\label{eq:isomorphismofgraded3}
U(\gr_{F_\mathbb{Z}}(L)) \cong \gr_{F_\mathbb{Z}}(U(L)).
\end{equation}
which induces a valuation $\upsilon\colon U(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ as in Section~\ref{sec:generalfiltrations}.
In what follows, the two
objects in \eqref{eq:isomorphismofgraded3} will be identified
via the isomorphism given in either \cite[Proposition~1]{Vergne} or
\cite[Lemma~2.1.2]{Boiscorpsenveloppants}. This isomorphism sends
the class of an element of $\mathcal{B}_n$ in $L_n$ to
its class in $U(L)_n$.
The valuation $\upsilon\colon U(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ can be
extended to a valuation $\upsilon\colon \mathfrak{D}(L)\rightarrow
\mathbb{Z}\cup\{\infty\}$ \cite[Proposition~9.1.1]{Cohnskew}.
Consider $u,v$ and $w=[v,u]$. Note that $\upsilon(u)=\upsilon(v)=-1$
and $\upsilon(w)=-2$ because $L$ is not two-dimensional. Denote by
$\bar{u}$, $\bar{v}$ the class of $u,v\in U(L)_{-1}$ and also the
class of $u$ and $v$
|
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] | 1,024 |
ArXiv
| 0.054076 | 2.362612 | -0.831938 | 0.428541 |
in $L_{-1}$. Denote by $\bar{w}$ the class of
$w$ in $U(L)_{-2}$ and in $L_{-2}$. By
Lemma~\ref{lem:gradedOre}(4), $U(\gr_{F_\mathbb{Z}}(L))$ is an Ore domain.
Let $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$ be its Ore ring of fractions.
Observe that $[\bar{v},\bar{u}]=\bar{w}$ as elements of
$\gr_{F_\mathbb{Z}}(L)$.
Define
$\overline{V}=\frac{1}{2}(\bar{u}\bar{v}+\bar{v}\bar{u})$, and consider the following
elements of $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$:
$$\overline{S}_1=\bar{w}^{-1}\Big((\overline{V}-\frac{1}{3}\bar{w})
(\overline{V}+\frac{1}{3}\bar{w})^{-1}+ (\overline{V}-\frac{1}{3}\bar{w})^{-1}(\overline{V}+\frac{1}{3}\bar{w})\Big) \bar{w}^{-1},$$
$$\overline{T}_1=(\bar{w}+\bar{v}^2)^{-1}(\bar{w}-\bar{v}^2)\overline{S}_1(\bar{w}+\bar{v}^2)(\bar{w}-\bar{v}^2)^{-1}.$$
By Corollary~\ref{coro:freesymmetricresiduallynilpotent}(3)(i)(b),
the $k$-subalgebra of $\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))$
generated by $\{\overline{S}_1\overline{S}_1^*, \overline{T}_1\overline{T}_1^*\}$ is the free
$k$-algebra on $\{\overline{S}_1\overline{S}_1^*, \overline{T}_2\overline{T}_2^*\}$.
Let $\mathcal{H}$ be the set of homogeneous elements of $\gr_{F_\mathbb{Z}}(U(L))$. From
\eqref{eq:isomorphismofgraded3} and Lemma~\ref{lem:gradedOre}, we
obtain the following commutative diagram
$$\xymatrix{\gr_{F_\mathbb{Z}}(U(L))\cong U(\gr_{F_\mathbb{Z}}(L))\ar@{^{(}->}[r]\ar@{^{(}->}[d]
&
\mathfrak{D}(\gr_{F_\mathbb{Z}}(L))\\
\mathcal{H}^{-1}\gr_{F_\mathbb{Z}}(U(L))\cong\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))
\ar@{^{(}->}[ur] & },$$ where the diagonal arrow is obtained from the
universal property of the Ore localization. Note that
$\overline{V},\, \overline{V}-\frac{1}{3}\bar{w},\,
\overline{V}+\frac{1}{3}\bar{w}$ are homogeneous elements of degree $-3$,
and the elements $\bar{w}+\bar{v}^2,\,
\bar{w}-\bar{v}^2$ are homogeneous elements of degree $-2$ in
$\gr_{F_\mathbb{Z}}(U(L))$. Thus $\overline{S}_1$,
$\overline{S}_1^*$, $\overline{T}_1$,
$\overline{T}_1^*$ are in fact homogeneous elements
of degree $-4$ in $\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$.
Now observe that $S_1$, $S_1^*$, $T_2$ and $T_2^*$ are elements of $\mathfrak{D}(L)$ such
that $\upsilon(S_1)=\upsilon(S_1^*)=\upsilon(T_2)=\upsilon(T_2^*)=4$,
hence $\upsilon(S_1S_1^*)=8$, $\upsilon(T_2T_2^*)=8$ and
$\overline{S}_1\overline{S}_1^*=S_1S_1^*+\mathfrak{D}(L)_{>8},\,
\overline{T}_2\overline{T}_2^*=T_2T_2^*+\mathfrak{D}(L)_{>8}$ in
$\gr_{F_\mathbb{Z}}(\mathfrak{D}(L))$.
|
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] | 1,024 |
ArXiv
| 0.161739 | 2.424906 | -0.733759 | 0.133824 |
Defining $A=1+S_1S_1^*$ and $B=1+T_1T_1^*$, the result follows from Theorem~\ref{coro:divisionrings}.
\end{proof}
As a corollary, we obtain a generalization of \cite[Theorem~5.2]{FerreiraGoncalvesSanchez2},
where the existence of a free $k$-algebra was proved. We recall that
the principal involution on a Lie $k$-algebra $L$ is defined by $L\rightarrow L$, $f\mapsto -f$.
\begin{coro}\label{coro:symmeticprincipalinvolution}
Let $k$ be a field of characteristic zero and $L$ be a Lie $k$-algebra such that its
universal enveloping algebra $U(L)$ is an Ore domain. Let
$\mathfrak{D}(L)$ be its Ore ring of fractions. Let $u,v\in L$ be such that
the Lie subalgebra generated by them is of dimension at least three.
Then there exist symmetric elements $A,B\in\mathfrak{D}(L)$ with respect
to the principal involution such that the $k$-subalgebra
generated by $\{A,A^{-1},B,B^{-1}\}$ is the free group $k$-algebra on $\{A,B\}$. \qed
\end{coro}
\section{Free group algebras in the Malcev-Neumann division ring of fractions
of a residually torsion-free nilpotent group}\label{sec:Heisenberggroup}
In this section, for a group $G$ and elements $x,y\in G$, $(y,x)$ denotes the
commutator $(y,x)=y^{-1}x^{-1}yx$. Also, if $A,B$ are subgroups of $G$, $(B,A)$
denotes the subgroup of $G$ generated by the commutators $(y,x)$ with $y\in B$, $x\in A$.
\medskip
Let $R$ be a ring and $(G,<)$ be an ordered group. Suppose that
$R[G]$ is the group ring of $G$ over $R$.
We define a new ring,
denoted $R((G;<))$ and called \emph{Malcev-Neumann
series ring}, in which $R[G]$ embeds. As a set,
\[
R((G;<))=\Bigl\{f=\sum_{x\in G}a_x x : a_x\in R,\ \supp(f)
\text{ is well-ordered}\Bigr\},
\] where $\supp(f)=\{x\in G\mid a_x\neq 0\}$.
Addition and multiplication are defined extending the ones in
$R[G]$. That is, given $f=\sum\limits_{x\in G}a_x x
$ and $g=\sum\limits_{x\in G}b_x x$, elements of
$R((G;<))$, one has
\[
f+g=\sum_{x\in G}(a_x+b_x) x \quad\text{and}\quad
fg=\sum_{x\in G}\Bigl(\sum_{yz=x}a_yb_z \Bigr)x.
\]
It was shown, independently, in \cite{Malcev}
and \cite{Neumann} that if
$R$ is a division ring, then
$R((G;<))$ is a division ring.
If $k$ is a field, the division subring of $k((G;<))$
generated by the group ring $k[G]$ will be
called the \emph{Malcev-Neumann division ring of fractions} of
$k[G]$ and it will be denoted by $k(G)$. It
is important to observe the following. For a subgroup $H$ of $G$,
$k((H;<))$ and $k(H)$ can be regarded
as division subrings of $k((G;<))$ and
$k(G)$, respectively, in the natural way. We remark that
$k(G)$ does not depend on the order $<$ of $G$,
see \cite{Hughes}. When the group ring $k[G]$
|
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] | 1,024 |
ArXiv
| 0.671868 | 2.362612 | 0.146714 | 1.14952 |
is
an Ore domain, then $k(G)$ is the Ore ring of fractions of $k[G]$.
\medskip
An \emph{involution} \label{involutiongroup}
on a group $G$ is a map $*\colon G\rightarrow G$, $x\mapsto x^*$,
that satisfies $$(xy)^*=y^*x^*\ \textrm{ and } \ (x^*)^*=x \textrm{ for all } x,y\in G.$$
In other words, $*$ is an anti-automorphism of order two.
Suppose that $G$ is a group endowed with an involution $*\colon G\rightarrow G$, $x\mapsto x^*$,
$k$ is a field and $k[G]$ is the group $k$-algebra. The map
$*\colon k[G]\rightarrow k[G]$ defined by $\Big(\sum\limits_{x\in G} a_xx\Big)^*=\sum\limits_{x\in G}
a_xx^*$ is a $k$-involution on $k[G]$.
If $(G,<)$ is an ordered group, we remark that the $k$-involution on the group algebra $k[G]$
induced the involution
$\ast$ on $G$ extends uniquely to a $k$-involution on the Malcev-Neumann
division ring of fractions $k(G)$ of $k[G]$,
see \cite[Theorem~2.9]{FerreiraGoncalvesSanchez1}.
\medskip
Let $G$ be a group. If $H$ is a subgroup of $G$, we denote by $\sqrt{H}$ the subset of $G$ defined by
\[\sqrt{H}=\{x\in G : x^n\in H \text{, for some } n\geq 1\}.\]
We shall denote the $n$-th
term of the lower central series of $G$ by $\gamma_n(G)$. That is,
we set $\gamma_1(G)=G$ and, for $n\ge 1$, define
$\gamma_{n+1}(G)=(G,\gamma_n(G))$.
A group $G$ is \emph{residually torsion-free nilpotent} if for
each $g\in G$, there exists a normal subgroup $N_g$ of $G$ such that
$g\notin N_g$ and $G/N_g$ is torsion-free nilpotent. Equivalently,
$\bigcap_{n\geq 1}\sqrt{\gamma_n(G)}=\{1\}$.
It is well known that any residually torsion-free nilpotent group is orderable, see
for example \cite[Theorem~IV.6]{Fuchs}.
Let $G$ be a residually torsion-free nilpotent group, $k$ be a field of characteristic zero
and consider the group algebra $k[G]$. Consider an involution on $G$
and its extension to the Malcev-Neumann
division ring of fractions $k(G)$ of $k[G]$.
The aim of this section is to prove that
there exist symmetric elements in $k(G)$ that generate a noncommutative
free group $k$-algebra. For that we will need the following discussion.
\bigskip
Let $G$ be a torsion-free nilpotent group. An \emph{$\mathcal{N}$-series} of $G$ is a sequence
$\{H_i\}_{i\geq 1}$,
$$G=H_1\supseteq H_2\supseteq\dotsb\supseteq H_n\supseteq\dotsb$$
of normal subgroups of $G$ that satisfies the following three conditions
$$(H_i,H_j)\subseteq H_{i+j},\quad \bigcap_{i\geq 1}H_i=\{1\},\quad G/H_i \textrm{ is torsion free for all } i\geq 1.$$
The $\mathcal{N}$-series induces a \emph{weight function} $w\colon G\rightarrow \mathbb{N}\cup\{\infty\}$ defined
by $w(g)=i$ if $g\in H_i\setminus H_{i+1}$ and $w(1)=\infty.$
Let $k$
|
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] | 1,024 |
ArXiv
| 0.564544 | 2.435288 | 0.281654 | 1.184937 |
be a field of characteristic zero, $G$ torsion-free nilpotent group
with an $\mathcal{N}$-series $\{H_i\}_{i\geq 1}$ and consider the group ring $k[G]$. The
$\mathcal{N}$-series defines the \emph{cannonical filtration}, $F_\mathbb{Z}k[G]=\{F_nk[G]\}_{n\in\mathbb{Z}}$,
induced by $\{H_i\}_{i\geq 1}$ which is defined by
$F_nk[G]=k[G]$, for all $n\leq 0$, and for $n\geq 1$, $F_nk[G]$ is the $k$-vector space
spanned by the set $$\Big\{(h_1-1)(h_2-1)\cdots(h_s-1)\colon s\geq 1,\ \sum_{j=1}^s w(h_j)\geq n\Big\}$$
Note that $F_1k[G]$ is the augmentation ideal of $k[G]$ and that $F_nk[G]\cdot F_mk[G]\subseteq F_{n+m}k[G]$.
For each $i\geq 1$, $H_i/H_{i+1}$ is an abelian group. Denote the operation
additively. More precisely, if $x_i,x_i'\in H_i$, $\widetilde{x_i}$ denotes the class $x_iH_{i+1}$.
Then $\widetilde{x_i}+\widetilde{x_i'}=\widetilde{x_ix_i'}$ in $H_i/H_{i+1}$.
The abelian group $L(G)=\bigoplus\limits_{i\geq 1}H_i/H_{i+1}$ can be endowed
with a $\mathbb{Z}$-graded
Lie $\mathbb{Z}$-algebra structure with the following product on homogeneous elements
$[\widetilde{x_i},\widetilde{x_j}]=\widetilde{(x_i,x_j)}\in H_{i+j}/H_{i+j+1}$,
for $x_i\in H_i$, $x_j\in H_j$, and then extending by bilinearity. Then
$k\otimes_\mathbb{Z}L(G)$ is a Lie $k$-algebra with universal enveloping algebra
$U(k\otimes_\mathbb{Z}L(G))$.
In \cite[Theorem~2.3]{LichtmanmatrixringsI},
Lichtman proved
a more general version of \cite{Quillengraded},
in a similar way as Quillen's result is shown in \cite[Chapter~VIII]{Passibookaugmentation}.
Lichtman's result implies that that
there exists an isomorphism of
$\mathbb{Z}$-graded $k$-algebras
\begin{eqnarray}\label{eq:isomorphismQuillen}
\Theta\colon U(k\otimes_\mathbb{Z}L(G)) &\rightarrow &\gr_{F_\mathbb{Z}}(k[G]) \\
\widetilde{x_i} & \mapsto &(x_i-1)+F_{i+1}k[G]. \nonumber
\end{eqnarray}
\bigskip
Let $\mathbb{H}=\langle a,b\mid (b,(b,a))=(a,(b,a))=1 \rangle$ be the Heisenberg group.
Define $c=(b,a)$.
Consider the following \emph{main involutions} of $\mathbb{H}$ which are defined on the generators and
extended accordingly.
\begin{enumerate}[(1)]
\item $a^*=a$, $b^*=b^{-1}$ and $c^*=c$.
\item $a^*=a$, $b^*=b$ and $c^*=c^{-1}$.
\item $a^*=a^{-1}$, $b^*=b^{-1}$ and $c^*=c^{-1}$.
\end{enumerate}
\begin{prop}\label{prop:symmetricHeisenberggroup}
Let $k$ be a field of characteristic zero.
Let $\mathbb{H}=\langle a,b\mid (b,(b,a))=(a,(b,a))=1 \rangle$ be the Heisenberg group and $c=(b,a
|
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] | 1,024 |
ArXiv
| 0.427332 | 2.632551 | 0.074537 | 0.696694 |
)$.
Consider the
group $k$-algebra $k[\mathbb{H}]$ and its Ore ring of fractions $k(\mathbb{H})$.
Consider the elements of $k(H)$
$$\mathcal{V}=\frac{1}{2}((a-1)(b-1)+(b-1)(a-1)),$$
$$\mathcal{S}_2=(c-1)\Big((\mathcal{V}-\frac{1}{3}(c-1))(\mathcal{V}+\frac{1}{3}(c-1))^{-1}+
(\mathcal{V}-\frac{1}{3}(c-1))^{-1}(\mathcal{V}+\frac{1}{3}(c-1))\Big)(c-1),$$
$$\mathcal{T}_3=((c-1)+(b-1)^2)^{-1}((c-1)-(b-1)^2) \mathcal{S}_2 ((c-1)+(b-1)^2)((c-1)-(b-1)^2)^{-1},$$
$$\mathcal{T}_4= ((c-1)^2+(b-1)^3)^{-1}((c-1)^2-(b-1)^3)\mathcal{S}_2((c-1)^2+(b-1)^3)((c-1)^2-(b-1)^3)^{-1}.$$
The following statements hold true.
\begin{enumerate}[\rm(1)]
\item Suppose that $*\colon \mathbb{H}\rightarrow \mathbb{H}$ is one of the main involutions (2) or (3) above.
Then the $k$-subalgebra of $k(\mathbb{H})$ generated by
$$\{1+\mathcal{S}_2\mathcal{S}_2^*,
(1+\mathcal{S}_2\mathcal{S}_2^*)^{-1}, (1+\mathcal{T}_3\mathcal{T}_3^*), (1+\mathcal{T}_3\mathcal{T}_3^*)^{-1} \}$$
is the free group $k$-algebra on the set $\{1+\mathcal{S}_2\mathcal{S}_2^*,
(1+\mathcal{T}_3\mathcal{T}_3^*)\}$.
\item Suppose that $*\colon \mathbb{H}\rightarrow \mathbb{H}$ is the main involution (1) above.
Then the $k$-subalgebra of $k(\mathbb{H})$ generated by
$$\{1+\mathcal{S}_2\mathcal{S}_2^*,
(1+\mathcal{S}_2\mathcal{S}_2^*)^{-1}, (1+\mathcal{T}_4\mathcal{T}_4^*), (1+\mathcal{T}_4\mathcal{T}_4^*)^{-1} \}$$
is the free group $k$-algebra on the set $\{1+\mathcal{S}_2\mathcal{S}_2^*,
(1+\mathcal{T}_4\mathcal{T}_4^*)\}$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Consider the following $\mathcal{N}$-series of $\mathbb{H}$.
$$H_1=\mathbb{H}\supseteq H_2=(c)\supseteq H_3=\{1\}.$$
If we set $x=aH_2,y=bH_2\in H_1/H_2$ and $z=cH_3\in H_2/H_3$,
then the $\mathbb{Z}$-graded Lie $\mathbb{Z}$-algebra $L(H)$ has as
$\mathbb{Z}$-basis the elements $x,y,z$ with products $[y,x]=z$, $[y,z]=[x,z]=0$.
Hence the $\mathbb{Z}$ graded Lie $k$-algebra $k\otimes_\mathbb{Z} L(\mathbb{H})$ is the
Heisenberg Lie $k$-algebra $H$ with the
$\mathbb{Z}$-grading given in Example~\ref{ex:gradedLie}(c).
The isomorphism \eqref{eq:isomorphismQuillen} implies that the cannonical filtration
induced by the $\mathcal{N}$-series is in fact a valuation, because the graded ring is
a domain. Since $k[\mathbb{H}]$ is an Ore domain, the valuation can be extended to
a valuation $\upsilon\colon k(\mathbb{H})\rightarrow \
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ArXiv
| -0.263141 | 2.113437 | -1.000501 | 0.173035 |
mathbb{Z}\cup\{\infty\}$.
If we let $\mathcal{H}$ be the homogeneous elements of $\gr_{F_\mathbb{Z}}(k[\mathbb{H}])$,
Lemma~\ref{lem:Reesgradedring}(3) implies that there exists an isomorphism
of $\mathbb{Z}$-graded $k$-algebras
\begin{equation}\label{eq:isoQuillenHeisenberg}
\gr_\upsilon(k(\mathbb{H}))\cong
\mathcal{H}^{-1}\gr_{F_\mathbb{Z}}(k[\mathbb{H}])\cong \mathcal{H}^{-1}U(k\otimes_\mathbb{Z}L(\mathbb{H})).
\end{equation}
Observe that $\mathcal{H}^{-1}U(k\otimes_\mathbb{Z}L(\mathbb{H}))\hookrightarrow
\mathfrak{D}(k\otimes_\mathbb{Z}L(\mathbb{H}))$. Now note that
$$\mathcal{V}, \mathcal{V}\pm(c-1), (c-1), (b-1)^2, (c-1)\pm(b-1)^2 \in F_2k[\mathbb{H}]\setminus F_3k[\mathbb{H}].$$
Hence the classes of these elements in $\gr_{F_\mathbb{Z}}(k[\mathbb{H}])$ are homogeneous of degree two.
It implies that the class of $\mathcal{S}_2$ and $\mathcal{T}_3$ in $\gr_{\upsilon}(k(\mathbb{H}))$
are homogeneous of degree four. Moreover, their image under the isomorphism \eqref{eq:isoQuillenHeisenberg}
are the elements $S_2,T_3\in \mathfrak{D}(k\otimes_\mathbb{Z}L(\mathbb{H}))$ given in
Theorem~\ref{theo:freegroupHeisenberg}(4).
Since each $H_i$ is invariant under the involution $*$, it induces a $k$-involution
in the Lie $k$-algebra $k\otimes_\mathbb{Z}L(\mathbb{H})$. Hence the isomorphism \eqref{eq:isoQuillenHeisenberg}
is an isomorphism of $*$-algebras, i.e. $\Phi(f^*)=\Phi(f)^*$.
Note that the induced involution on $k\otimes_\mathbb{Z}L(\mathbb{H})$ is one of the
involutions in Lemma~\ref{lem:equivalentinvolutionHeisenbergalgebra}(2) or (3). By
Theorem~\ref{theo:freegroupHeisenberg}(4)(a), the elements $S_2$, $T_2$ are symmetric with respect to
the induced involution on $\mathfrak{D}(k\otimes_\mathbb{Z}L(\mathbb{H}))$.
Hence, the image of the classes of $\mathcal{S}_2^*$ and $\mathcal{T}_3^*$ are also $S_2$
and $T_3$, respectively. The classes
of the elements $\mathcal{S}_2\mathcal{S}_2^*, \mathcal{T}_3\mathcal{T}_3^*$
in $\gr_{\upsilon}(k(\mathbb{H}))$
are homogeneous of degree $8$. Moreover, they generate a free algebra in
$\gr_{\upsilon}(k(\mathbb{H}))$, because $S_2^2$ and $T_3^2$ generate a free algebra
in $\mathfrak{D}(k\otimes_k L(\mathbb{H}))$ by Theorem~\ref{theo:freegroupHeisenberg}(4)(b).
Now the result follows by Theorem~\ref{coro:divisionrings}.
(2) It follows in the same way as (1). Now one has to consider the $\mathcal{N}$-series
$$H_1=G\supseteq H_2=\langle b,c\rangle\supseteq H_3=\langle c\rangle\supseteq H_4=\{1\}.$$
Then again $k\otimes_\mathbb{Z} L(H)$ is the Heisenberg Lie $k$-algebra, but with the gradation
given in Example~\ref{ex:gradedLie}(4). Then
the isomorphism in \eqref{eq:isoQuillenHeisenberg} (with a
different gradation) sends $\mathcal{S}_2$ and $\mathcal{T}_4$
to the elements $S
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ArXiv
| 0.307781 | 2.622169 | -0.63558 | 0.403243 |
International Ihc 9400i Eagle Service Manuals - May 27, 2009 Β· International 9400i Eagle Semi Dash Tour BlackwolfTrucker. Loading Unsubscribe from BlackwolfTrucker? D.O.T put me out of servicesmh - Duration: 5:19.. Search Results for International 9400i Trucks. More than 5 Results Found. All (5) Trucks (5) Trailers (0) Parts (0) Eagle, (6) Alum. Wheels, Very Clean. More Information. Model: 9400i . 2007 IHC 9400I, ISX Cummins Engine, Engine Brake, FROF 16210C Transmission, More Information. 2005 International 9400i. Call for Price. International 9400 Owners S Manual Read/Download truck 05 owner manual pdf international truck code scanner schedule international truck service manual international harvester truck. Find Recipes Β· Owner's Resources Β· Product Support Β· Repair & Owners Antique Tractor Manual Shop. 2001 international 9400 eagle 2007 9400.
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|
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] | 1,024 |
C4
| -2.22452 | -1.733197 | -1.494534 | -1.779934 |
Racino
VLTs
Big Six Wheel
Website: https://www.pragmaticplay.com/en/
π Who Are Pragmatic Play?
π History of Pragmatic Play
π Who Owns Pragmatic Play?
π How Do You Contact Pragmatic Play?
π What Are Pragmatic Play's Most Popular Slots?
π How Much is Pragmatic Play Worth?
Pragmatic Play is a renowned developer in the iGaming industry. The company provides a wide array of slots, live casino games, and bingo. Pragmatic Play has an objective for providing players with quality entertainment. Licensed by Malta, Curaçao, and Romania, the developer is certified in over 20 jurisdictions. Based in Malta, the company has offices in Gibraltar, the UK, India, Ukraine, and the Philippines.
The games offered by Pragmatic Play are available in more than 30 languages and all currencies. Also, the games can be accessed via a mobile phone or desktop. They are compatible with iOS and Android devices. The games can also be downloaded, which is a plus for players with slow internet.
Additionally, independent gaming regulation bodies always assess games offered by the developer to ensure they are fair and of integrity. The company uses third parties like Technical Services Bureau (TSB) and Gaming Laboratories International (GLI) to test their games extensively before they are introduced to the market. Furthermore, this multi-award-winning content developer is appreciated by players from all over the world due to their innovative entertainment products.
Pragmatic Play was founded in 2007 under the name TopGame Technology. In 2008, the developer launched its first game. Over a couple of years, TopGame Technology became one of the most popular developers. In 2015, the company decided to rebrand, hence it changed its name to Pragmatic Play. Moreover, it updated all its software using HTML 5, which put it on the map in the gaming market. In 2016, it was licensed by Curacao and has since brought amazing games into the industry.
Pragmatic Play is owned by IBID Group, a group of European investors, who acquired it on July 22, 2016. However, the company is managed by experts from all over the world. The current CEO is Julian Jarvis, a position he has held since January 2020. Julian also serves as the Group CEO and Chief Legal Officer of IBID Group.
You can contact Pragmatic Play in several ways. First, you can write your message on the website, and the company will send you an email. You can also send a direct email through [email protected], and the team will reply to you on time.
Moreover, if you are near the company's offices, you can visit them. The addresses of a few offices are 144 Tower Road, Sliema, SLM1604, Malta, Gordon House, 10a Prospect Hill, Douglas, IM1 1EJ, Isle of Man, and Office 1.22, World Trade Center, 6 Bayside Road, Gibraltar, GX11 1AA.
Pragmatic Play offers over 150 slots and 80 table games. Some of them are incredibly popular. Examples of their popular slots are :
Besides these slots, Pragmatic Play has progressive slots that give players a chance for higher prizes. For example, Wolf Gold has three progressive slots, which are mini jackpots seed at 750 coins, major jackpots seed at 2500 coins, mega jackpots seed at 25,000 coins. Also, most of the slots have reliable RTPs, usually ranging from 96% to 96.6%.
Examples of table games offered by Pragmatic Play are blackjack, roulette, dragon tiger, and baccarat. The developer also offers live dealer games, bingo, drops and wins, and virtual sports.
Currently, Pragmatic Play is worth $64 million. This growth is due to the positive reception the developer has received from players all over the world. The company has partnered with hundreds of casino sites to offer its games. This includes 888 Casino, King Bills, Bit Starz, Playamo,
|
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] | 1,024 |
CommonCrawl
| -0.195215 | -1.106367 | 0.452907 | -0.368806 |
Q: Java generics casting to > I have the following class
public class DBField<T>
{
protected String fieldName;
protected FieldConverter c;
protected T value;
protected DataObject dataObject;
public T getValue()
{
return value;
}
public void setValue(T value)
{
this.value = value;
}
public DBField(DataObject dataObject, String fieldName, FieldConverter c)
{
this.fieldName = fieldName;
this.c = c;
this.dataObject = dataObject;
}
}
T is supposed to be Boolean, Float, String etc.
protected void ValuesToFields(List<Object> values, List<DBField<?>> fields) throws Exception
{
if (values.size() != fields.size())
throw new Exception("Length does not match.");
for (int i = 0; i < values.size(); i++)
{
Class valueClass = values.get(i).getClass();
Class fieldClass = fields.get(i).getValue().getClass();
if (valueClass.equals(fieldClass))
{
fields.get(i).setValue(values.get(i));
}
else
throw new Exception("type mismatch");
}
}
Object is also supposed to contain Boolean, Float, String etc.
The problem with this code is
fields.get(i).setValue(values.get(i));
The syntax checker tells me I need to cast values.get(i) (to ? i suspect). How do I do this? I already tried valueClass.cast(values.get(i)) but no luck.
A: In order for your code to be safe, for each i, the i'th element of values must be an instance of the type parameter of the DBField that is the i'th element of fields. Your code does not guarantee that this holds, and in fact there is no way to declare them in Java to ensure that this relationship between corresponding elements is true. And due to type erasure, you can't even check at runtime that the elements are right, because given a field, you don't know its type parameter. So there must be some unchecked casts, and we must take on faith that the arguments are correct.
The simplest thing to do would be to cast each field to DBField<Object>:
((DBField<Object>)fields.get(i)).setValue(values.get(i));
This is kind of saying "trust us, we know that this field can take any Object", and thus it can take a value of any type. It is kind of lying, because we know there are supposed to be fields whose type parameter is not Object, but since we must make some kind of unchecked cast anyway, this "unsafe cast" is no worse than the other solutions.
Alternately, if you don't want to do this arguably dubious cast, a more "legitimate" way would be to write a private helper method -- a "wrapper helper" -- which explicitly names the type parameter of the field, allowing us to simply cast to the value to this type:
private <T> static void ValueToField(Object value, DBField<T> field) {
field.setValue((T)value);
}
//...
ValueToField(values.get(i), fields.get(i));
Note that the cast here is also an unchecked cast. The disadvantage of this method is that it requires the overhead of writing an extra method.
P.S. Your checks with valueClass and fieldClass are not very good. First of all, if the value of a field is currently null, it will cause a null pointer exception. Also, the value of a DBField<T> is any instance of T, whose actual class may be a subclass of T; so if you use this to check, it might lead to bad results. It's probably best if DBField contains the class object of the class of T, so it can be used to check. Also, you shouldn't compare equality with the value's actual class, since a subclass of T would also work, so you should check fieldClass.isInstance(values.get(i)) instead.
A: if(values.get(i) instanceof valueClass) ?
A: You have a List<Object>.
You should have a List<DBField<Object>> to match it, or change your first list to List<?>
|
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StackExchange
| -0.746438 | 0.935049 | -0.623027 | 0.886426 |
HomeΒ»Marley CoffeeΒ»Marley Coffee Comes to 1stinCoffee.com Mon!
Marley Coffee Comes to 1stinCoffee.com Mon!
Extremely reflective, Bob Marley constantly looked at the world and at himself with the desire to make a difference. It came through in his music, which in so many ways is a poetic call to every individual to shape themselves and the world they've inherited into something better. His lyrics worked to instill a sense of pride and strength in the face of injustice and adversity.
An international success and cross-cultural inspiration, Marley began his life humbly in the rural areas of Jamaica. It was his hope to eventually return to his roots and work the land. Marley's son, Rohan, has worked to combine his father's philosophy and love for farming by co-founding Marley's Coffee with his friend Shane Whittle.
Marley's Coffee is an organic coffee company that strives to support local communities and the environment by adhering to environmentally sustainable and socially ethical practices. It has been said that "β¦intellectual activity has to be informed by the arts of the imaginationβ¦" It is by working to help the people: the farmers and children of the local communities, that Rohan and Whittle have honored Bob Marley's fierce desire to better the world. By working solely with like-minded free-trade farmers in Indonesia, Ethiopia, Jamaica and Central America, Marley's Coffee remains at the forefront of sustainable development.
They have emulated not only Marley's humanitarian beliefs, but his sense of pride as well. Each blend of coffee (named after Bob Marley songs) is completely organic and offers delightful, rich bouquets that rank up with some of the finest coffees in the world.
"Lively Up", named after Marley's inspirational song, Rastaman, Lively Up, is a 5-bean espresso blend that works in perfect harmony with premier Ethiopian Yirgacheffe. Full-bodied and well-balanced, the coffee has a long and savory finish with hints of candied fruit, sweet caramel and cocoa. "Lively Up" can be prepared using either drip, press or espresso methods.
"One Love", a tilt-of-the-hat to one of Marley's most widely acclaimed songs, is a well-rounded, brightly acidic coffee made of 100% Ethiopian Yirgacheffe. "One Love" is one of Marley's Coffee's most popular blends, defined by floral notes and hints of blueberry, cocoa and spices. This cup will finish off with a profound cherry finish.
"Mystic Morning" is a blend of Ethiopian Yirgacheffe and the finest grade 1 coffee beans from Central America. A medium-bodied blend, "Mystic Morning" is known for its vibrant spice flavors and undertones of cocoa and cinnamon.
"Buffalo Soldier" is a blend of Ethiopian Yirgacheffe and the finest beans from Central America and Indonesia. Sweetly roasted with delicious smoked earth notes, this sultry coffee blend has dark chocolate and berry undertones with a smooth and lingering finish.
Coffee ITAL seal and is recognized the world-over by coffee connoisseurs as one of the best coffees on the market.
Marley's coffee also offers a delicious decaf blend called "Simmer Down." This distinctive coffee is made with a unique process called the "Swiss Water" technique. Pure water from the mountains of British Columbia to gently removes the caffeine. It is in this way that the beans maintain their delicate flavors and smooth characteristics. "Simmer Down" is roasted dark and full-bodied with mild acidity and subtle aromas of nutmeg, spices and cocoa. Marley's Coffee also offers a decaffeinated blend called "Mountain Roast Decaf" which is also made using the Swish Water technique.
Now, you can enjoy each of Marley's Coffee's rich organic blends by ordering them online at http://www.1stincoffee.
|
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C4
| 1.299509 | -0.734633 | 0.273584 | -0.051164 |
The editors at LitHub recently wrote "cli-fi may very well end up being the defining literary genre of our era," in a listicle on cli-fi books. Well I have exciting news: This could be the world's first ELA cli-fi PBL! Designed for Secondary students of English Language Art, I have developed a sample Project Based Learning (PBL) that brings the literary genre of Climate Fiction to life for high school students.
Climate Fiction: What happens when Post-Apocalyptic Fiction moves to the Current Affairs aisle at Barnes and Noble.
Literature is an ever-evolving art form, which changes with the tastes of the time. To really get a sense of how marketing and literature intersect, it helps to examine genre, one of publishers' biggest marketing allies. This project plunges into an emerging genre known as cli-fi, enabling students to partake in the game of literary marketing by writing within a genre and then planning a strategy to attract readers to their own work. Please find and adapt my plan, goals, tips on monitoring and tracking students, and rubrics below.
Students explore a new genre, Climate Fiction and determine what classifies it as a genre.
Students explore how this genre is used for marketing purposes and create their own short story based on the tenets of this genre.
Students meet all of the requirements for the short story form by developing their story through careful planning, peer and teacher review, and revision.
Students then publish their stories and create self-directed marketing campaigns that drive students from class, teachers, parents, and other acquaintances to the site to read the story and comment on it (to prove they have read it).
Students present the documented results of their projects and vote on the best marketing campaigns in three different categories (Innovative Strategy, Creative Direction, and Impacting Message).
Finally, students reflect on the themes of the PBL and write reflections, essays, and feedback on their learning experience.
Collaborate in discussions and peer revision activities to support one another's learning.
Develop skills in creative writing and revision.
Explore and interpret how genre and theme influence the marketing of a work of literature.
Produce a self-directed marketing campaign for your own literary work and give a presentation on this process to the class.
A note on Monitoring and Data Collection: Below, you'll find an overview of Monitoring (in bold) and Feedback (underlined) activities along with rubrics for unit. With these assessments, I will monitor for activity completion, breadth and depth, willingness to develop and modify ideas, and citation of formal or personal examples/evidence in terms of emerging (1), developing (2), proficient (3), and advanced (4) ratings in a google spreadsheet, which students will have shared access to for their own grades. Students will also use the same general rating principles for peer and self reviews.
The rubrics are designed in a tiered structure, so that there are three main benchmarks after week 1, 3, and 5. Students will be given a final grade that is an average of all benchmark rubrics plus the collaboration and participation rubric.
Day One: Introduction to the project plan, rubrics, and resolve any questions.
Day Two: Define guiding question for Week 1: What is genre, and does "CliFi" satisfy the requirements of a genre?
Students brainstorm ideas about what makes a genre a genre by thinking of examples such as science fiction, mystery, and romance in a think-pair-share activity. Teacher reviews by giving verbal feedback and typing notes of the discussion to display on the board.
Students write a two paragraph definition of genre in their terminology notebooks, drawing from both their own explorations and by referencing definitions found through internet research, using their phones/computers. They turn this in at the end of class along with a KWL exit ticket on genre.
Homework: Students read selected short story examples from the CliFi genre as homework and discuss the next day.
Teacher reviews the KWL and starts class with a feedback discussion.
Students discuss how the examples they read do or do not conform to the genre in a Jigsaw-style literature circle activity that first takes place in groups with the same text, and the second time with individuals with all different texts. Teacher reviews by monitoring and taking notes of the classroom discussions and presenting key takeaways after the activity.
Students complete a
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| 0.929989 | 0.514566 | 0.538533 | 2.68761 |
think-pair-share by writing statements in Socrative that define CliFi. The statements are displayed to the whole class, once everyone has finished.
Students peer-review this list develop a list of key features of the genre by voting and discussing the most relevant features that are typed in a google doc using evidence based on their literary research about genre.
For Homework, suggest their edits using comments to the google doc and adding cited evidence that justify whether each statement is or isn't a valid genre definition. Teacher reviews these comments and gives feedback the next day.
In pairs, students design artistic posters that display these statements and their justifications to hang around the room. Students peer review the posters and give feedback to two other groups in the classroom discussion group.
Students discuss in groups of four how the CliFi genre is used as a marketing tool to sell books and present examples using books found online and their publishers' marketing messages.
Students create a shared google slide show of these examples, by taking them from the web. The slides should include a book, a sample text from its marketing material (a book review or publisher blurb), and a paragraph that describes why it falls within the CliFi genre. Teacher reviews and asks for revision for any that are not complete.
Students develop a fictional story concept related to Climate Change in some way as homework. This concept should be a general overview, like a film pitch, two paragraphs in length. These should be turned in the following day.
Students share and give feedback on the verbally shared story concepts of other students. Students revise and modify their story concepts using tracked changes on the based on feedback and turn in a second draft to use as a guide for themselves later on.
Homework: Students reflect in reflection journals (self-review) on how collaboration can improve the creative process.
Most of the rubrics are specific to the particular week of study, but the collaboration and participation rubric below applies to all weeks.
Guiding Question: What makes a good story?
Students brainstorm in groups what makes a good story and share their opinions in a mind-map format online.
I'll Use elements of Lajos Egri's "Art of Creative Writing" as a template. Students complete a planning worksheet about their stories. Teacher reviews this worksheet with written comments and suggestions for revision. Students will revise their work.
Students use a Graphic Organizer to develop the story further determine the: Setting, Main Characters, Point of View, Conflict, Plot, and Theme.
Students peer review each other's planning worksheets and give advice, ensuring that Climate Change factors as a central theme in the story.
Students brainstorm in a classroom discussion the most crucial English style and formatting conventions for writing fiction including proper dialogue punctuation and general paragraph length, etc. to review general good writing principles.
Teacher provides verbal feedback. A classroom note taker records the discussion for students to refer back to. (Other resources will be available as well).
Students write a first draft of the story by Monday of the next week.
Guiding Question: How can a good story become even better?
Review of story craft elements in group discussions.
Revision 1: Students complete a peer revision of another student's story based on elements from the plan in week 2: Setting, Main Characters, Point of View, Conflict, Plot, and Theme. Students see how well these elements are developed students annotate their feedback using the Diigo annotation tool.
Teacher review: the teacher monitors the work and adds comments to these reviews. All stories are expected to conform to English style and formatting conventions.
Students revise their story (over the weekend) and turn in the final on Monday.
All student stories are published on a class blog.
Students reflect on their strengths and weaknesses in developing a short story based on a genre.
Guiding Question: How can I get people to read my story?
Students choose a promotional project for their story such as Book Trailer, Poster, Social Media Hashtag Campaign, Email Marketing, or Word of Mouth Campaign (presentation). It can also include a combination of different elements.
Students watch a book promotion how to video for inspiration.
Students determine their primary message (step 1 from the video).
Students design their own rubric for the following criteria that they will vote on: Innovative Strategy, Creative Direction, and Impacting Message.
|
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C4
| 0.21676 | 0.30173 | 0.225167 | 2.728087 |
By Catherine Harrington , Madison Weaver on Wednesday, April 17th, 2019 at 2:32 p.m.
Trump is threatening to overturn the entire Affordable Care Act that provided over 200,000 West Virginians with healthcare coverage. Our Seniors depend on it for affordable prescriptions and pre-existing condition coverage.
Published: Wednesday, April 17th, 2019 at 2:32 p.m.
We want to hear your suggestions and comments. Email the Truth-O-Meter with feedback and with claims you'd like to see checked. If you send us a comment, we'll assume you don't mind us publishing it unless you tell us otherwise.
Carter said the United States is "the most warlike nation in the history of the world" due to a desire to impose American values on other countries, and he suggested that China is investing its resources into projects such as high-speed railroads instead of defense spending.
"We have wasted, I think, $3 trillion," Carter said, referring to American military spending. "China has not wasted a single penny on war, and that's why they're ahead of us. In almost every way.
"And I think the difference is if you take $3 trillion and put it in American infrastructure, you'd probably have $2 trillion left over. We'd have high-speed railroad. We'd have bridges that aren't collapsing. We'd have roads that are maintained properly. Our education system would be as good as that of, say, South Korea or Hong Kong.
The former president said he understands that Trump is worried about China surpassing the U.S. as the world's top economic superpower.
"I don't really fear that time, but it bothers President Trump, and I don't know why. I'm not criticizing him β this morning," Carter said to laughs from the audience.
"President Jimmy Carter wrote President Trump a beautiful letter about the current negotiations with China and on Saturday they had a very good telephone conversation about President Trump's stance on trade with China and numerous other topics," said the statement, which wasn't attributed to a spokesperson.
Much of Carter's Palm Sunday lesson was focused on peace and kindness and was given before an audience that was mostly composed of visitors, many of whom had lined up overnight for the service.
Last month, Carter became the nation's longest-living president.
Stories like these are made possible by contributions from readers and listeners like you.
An industry lobbying offensive has put it on the cusp of achieving its holy grail: access to the resource-rich eastern Gulf of Mexico. The idea is so politically toxic in Florida that past presidents haven't even entertained it. But behind the scenes, oil and gas interests are appealing to Trump's desire to turbocharge U.S. energy production, including his past openness to drilling off the Florida coast.
The president and his top advisers haven't yet weighed in on the plan taking shape inside his Interior Department. But giving it the green light would be tantamount to a declaration of war on his second home state, given the uniform opposition from Florida Republicans, including prominent allies like Sen. Rick Scott, Gov. Ron DeSantis and others.
Industry representatives have said a plan has been imminent since last fall, but many expect the Interior Department is waiting for the Senate to confirm acting Secretary David Bernhardt to fill the agency's top slot before formally releasing the draft. Senate Majority Leader Mitch McConnell filed cloture Monday on Bernhardt's confirmation, teeing up a vote this week.
Our must-read briefing on what's hot, crazy or shady about politics in the Sunshine State.
The administration's position was muddied when former Interior Secretary Ryan Zinke held an elaborately staged Jan. 2018 meeting with Scott, then Florida's governor, to declare the state wouldn't be on the drilling map. The announcement was seen as a favor to boost Scott's electoral fortunes in his ultimately successful challenge against Democratic incumbent Bill Nelson, who tried to use environmental issues to separate himself from the Republican challenger.
People walk by election campaign billboards showing Israeli Prime Minister and head of the Likud party Benjamin Netanyahu
|
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C4
| 0.023849 | -0.687345 | -0.116442 | -0.530918 |
(left) alongside the Blue and White party leaders, including Benny Gantz. Ahead of Tuesday's election, Netanyahu has pledged to annex Israeli settlements in the occupied West Bank.
Netanyahu was asked on Israeli Channel 12 TV why he hasn't annexed Israeli settlement blocs in occupied territory, as NPR's Daniel Estrin reports from Jerusalem.
"If Netanyahu wants to declare Israeli sovereignty over the West Bank, then you know he has to face a real problem, the presence of 4.5 million Palestinians, what to do with them," Malki told the AP while attending the World Economic forum in Jordan, apparently citing the combined total of Palestinians living in the occupied West Bank, East Jerusalem and the Gaza Strip.
The Israeli settlements β which include large subdivisions and cities full of middle-class villas β have long complicated efforts for a two-state solution: Palestinians have said the settlements would make it impossible to create a viable state in the West Bank, as NPR's Greg Myre has reported.
Saeb Erekat, chief negotiator for the Palestine Liberation Organization, criticized Netanyahu's statement on Saturday.
Netanyahu's political campaign has emphasized his close ties with President Trump, Estrin reports. In his prime time interview on Saturday, Netanyahu portrayed those moves of support from the Trump administration as his own achievements, the AP reports.
Last month, Trump recognized Israeli sovereignty over the Golan Heights, which Israel seized from Syria in 1967. In his first year in office, Trump had also recognized Jerusalem β the disputed city claimed as capital by both Israeli and Palestinian people β as Israel's capital, breaking with decades of U.S. foreign policy.
Polls indicate a close race, though Netanyahu's Likud Party and its traditional allies, smaller right-wing parties, are predicted to win a slight majority of the votes. That gives Netanyahu the edge on forming a ruling coalition over Gantz's Blue and White political alliance β unless some right-wing parties choose to side with Gantz, Estrin reports.
Gantz has accused Netanyahu of inciting against Israel's Palestinian Arab citizens and embracing extremists by allying with the far-right Jewish Power Party.
Last year the US stopped contributing to the UN Relief and Works Agency(Unrwa), which has been looking after Palestinian refugees since 1949.
BEIJING β Chinese diplomats have been informed of the arrest of a Chinese woman at President Donald Trump's Mar-a-Lago club over the weekend and are providing her with consular services, the Foreign Ministry said Thursday.
Agent Samuel Ivanovich wrote in court documents that Zhang told him that she was there for a Chinese American event and had come early to familiarize herself with the club and take photos, contradicting what she had said at the checkpoint. He said Zhang said she had traveled from Shanghai to attend the nonexistent Mar-a-Lago event on the invitation of an acquaintance named "Charles," whom she only knew through a Chinese social media app.
Archived images of the United Nations Chinese Friendship Association website, which has since been taken down, show that the organization advertised itself as a non-profit registered with the U.N. Department of Economic and Social Affairs. A page of "registration documents" purports to show certificates from the States of Delaware and New York, as well as a screenshot of a listing on the U.N.'s official website.
But a search Thursday for the association on the U.N.'s database did not turn up any results.
The United Nations Chinese Friendship Association's website also shows Lee in photos with several government officials of various countries, including Trump, U.N. Secretary-General Antonio Guterres, as well as officials from China, Canada, Turkey and South Korea. It is not clear whether any of the photos have been digitally altered.
While no espionage charges have been filed against Zhang, her arrest has reignited concerns especially among Democrats that Trump's use of the club constitutes a security risk as long as members and guests are allowed to come in and out while he is there.
Zhang's arrest attracted comments from Chinese internet users on the popular Weibo microblogging service, many of whom portrayed her as having been tricked by those seeking to exploit her desire for attention
|
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e-Publications@Marquette
Home > Health Sciences > Biomedical Sciences, Department of > BIOMEDSCI_FAC > 227
Biomedical Sciences Faculty Research and Publications
Fibrinogen Birmingham: A Heterozygous Dysfibrinogenemia (AΞ± 16 Arg β His) Containing Heterodimeric Molecules
Kevin R. Siebenlist, Marquette UniversityFollow
J. T. Prchal, University of Wisconsin Medical School
Michael W. Mosesson, Blood Center of WisconsinFollow
Source ISSN
Original Item ID
DOI: 10.1182/blood.V71.3.613.613
Fibrinogen was isolated from the plasma of a 25-year-old female with a history of mild bleeding and several recent moderate to severe hemorrhagic episodes. Coagulability with thrombin approached 100% and varied directly with the time of incubation with the enzyme. High- performance liquid chromatography analysis of thrombin-induced fibrinopeptide release demonstrated retarded fibrinopeptide A (FPA) and fibrinopeptide B (FPB) release and the presence of an abnormal A peptide (FPA) amounting to 50% of the total. The same biochemical abnormalities were found in her asymptomatic father. Amino acid analysis and carboxypeptidase digestion of FPA demonstrated the substitution of His for Arg at A alpha 16. In contrast to the thrombin- and reptilase-sensitive Arg-Gly bond in the normal A alpha chain, the abnormal A alpha chain (A alpha) sequence is resistant to reptilase attack but is slowly cleaved by thrombin. To evaluate whether Birmingham A alpha and A alpha chains had been assembled nonselectively into heterodimeric (ie, 50% A alpha, A alpha) and homodimeric (ie, 25% A alpha, A alpha; 25% A alpha, A alpha) species, the clot and the clot liquor resulting from reptilase treatment of normal or Birmingham fibrinogen were separated, and each was then further incubated with thrombin to release remaining fibrinopeptides. Assuming that fibrinogen Birmingham contained heterodimeric molecules and that these and the normal molecules were completely incorporated into a reptilase clot, the expected coagulability would be 75%. In addition, subsequent thrombin treatment of the reptilase clot would release 50% of the total FPA and 75% of the total FPB present in the original sample. On the other hand, if only homodimeric fibrinogen species (50% A alpha, A alpha; 50% A alpha, A alpha) existed, the maximum reptilase coagulability would be 50%, and after thrombin treatment, 50% of the total FPB and no FPA would be recovered from the reptilase clot. We found the propositus's fibrinogen to be 68% coagulable, and we recovered 45% of the FPA and 70% of the FPB from the reptilase clot. Essentially the same coagulability and distribution of fibrinopeptides was found in the reptilase clot from her father's fibrinogen. We therefore conclude that fibrinogen Birmingham contains heterodimeric species (A alpha, A alpha) amounting to approximately 50% of the circulating fibrinogen molecules. The existence of heterodimers is consistent with a nonselective intracellular process of constituent chain assembly of dimeric plasma fibrinogen molecules.
Accepted version. Blood, Vol. 71, No. 3 (March 1, 1988): 613-618. DOI. Β© 1988 American Society of Hematology. Used with permission.
Siebenlist, Kevin R.; Prchal, J. T.; and Mosesson, Michael W., "Fibrinogen Birmingham: A Heterozygous Dysfibrinogenemia (AΞ± 16 Arg β His) Containing Heterodimeric Molecules" (1988). Biomedical Sciences
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Mount Umunhum: Open space agency votes to useβ¦
NewsCalifornia News
Mount Umunhum: Open space agency votes to use eminent domain to acquire road to top of former Air Force base
PUBLISHED: December 10, 2015 at 12:28 p.m. | UPDATED: March 10, 2017 at 5:25 p.m.
A Bay Area open space district, as expected, has voted to use eminent domain for the first time in nearly 20 years to acquire two pieces of land it needs to open public access to the top of Mount Umunhum, a former Air Force base in the hills south of San Jose.
By a 7-0 vote, the board of the Midpeninsula Regional Open Space District decided Wednesday night to file paperwork in court to forcibly acquire a 40-foot-wide strip of land running along about 1.5 miles of Mount Umunhum Road in the hills near Los Gatos from the McQueen family, which has owned it for roughly 60 years.
The agency also decided to forcibly purchase a public easement over a 200-yard-long section of the same road that crosses property owned by Mike Rossetta, of Los Gatos, and his brother, Leonard Rossetta, along with 19 other adjacent acres that the Rossettas own.
District officials β who say they have negotiated with the owners for three years without luck β propose to pay the McQueens $380,000, and the Rossettas $452,225. Under the law, if either disputes the district's appraised value, they can hire their own appraiser and take the case to a judge and jury to decide the sale price.
The 3,486-foot Mount Umunhum was home to the Almaden Air Force Station from 1957 to 1980. Radar crews operated on its summit, looking for Soviet bombers. But the base closed after satellites made its radar obsolete. In 1986, the open space district purchased the summit, left it padlocked for years and in 2013 tore down its dilapidated buildings, leaving only the distinctive concrete radar tower building.
The district, a public agency based in Los Altos, plans to open the summit to the public next October. It says the mountain, where it has purchased 18,000 acres over the past 30 years, could be a landmark for horse riders, bicyclists and hikers, similar to Mount Tamalpais or Mount Diablo.
Paul Rogers covers resources and environmental issues. Contact him at 408-920-5045. Follow him at Twitter.com/PaulRogersSJMN.
Paul Rogers has covered a wide range of issues for The Mercury News since 1989, including water, oceans, energy, logging, parks, endangered species, toxics and climate change. He also has worked as managing editor of the Science team at KQED, the PBS and NPR station in San Francisco, and has taught science writing at UC Berkeley and UC Santa Cruz.
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CommonCrawl
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Is FIFA 22 on Xbox Game Pass? Here are the FIFA games available on the service in 2022
7 free ways to entertain your kids in Portsmouth this weekend
The summer holidays are coming to an end and Portsmouth parent's might have run out of ideas to entertain their kids.Γ
Saturday, 1st September 2018, 1:06 pm
Updated Tuesday, 4th September 2018, 5:15 am
A view of Portsdown Hill from Portsmouth Harbour. Picture: Sarah Standing
If you've been to the beach too many times already this summer and don't want to spend a fortune '"Γ after buying all the back to school gear '"Γ we've rounded up seven free activities in Portsmouth.Γ
With the weather set to be sunny and warm again this weekend, you might fancy a final family adventure or day out before the new term begins.Γ
Never fear because we have rounded up seven great free activities to do in Portsmouth this weekend which will keep the whole family occupied.
Walk Portsdown Hill
Portsdown Hill has spectacular views across Portsmouth and the harbours, as well as the Isle of Wight.
It has an open network of paths which take walkers through wild-life rich habitats and flowery grassland, including birds such as kestrels.
There are also a number of forts which were built to defend the port from attack and Fort Nelson has been restored as an artillery museum which is free to enter.
Making it the perfect place for your kids to burn off their energy!Γ
Read More:Γ Portsmouth remains a magnet for overseas visitors as tourism booms i...
Visit Southsea Castle
Built in 1544, right by the sea, the castle was one of the fortifications built by Henry VIII to protect the coast from invaders.
However now it is open to the public for free.
Between March and October, visitors are able to explore the historic Southsea Castle, which is home to a cafe, gift shop and micro-brewery.
In the 19th Century a tunnel was built to defend the castle moat.
Visitors can still enter the tunnel and see how the castle would have been defended against invaders.
Artillery, gun platforms and panoramic views from the top of the keep overlook the Isle of Wight and Solent.
Who says you can't do a bit of learning while school is out.Γ
The cafe also offers a Champagne bar on Friday evenings '"Γ after all taking care of your children is thirsty work!Γ
Explore Portsmouth City Museum
It is free entry to Portsmouth City Museum for visitors, although donations are welcome.
Portsmouth City Museum is the 'Λmuseum of and for the people of Portsmouth' and features a variety of exciting displays including the Story of Portsmouth - which has reconstructions of a number of rooms from the cities history, including a 17th century bedchamber. There is also A Study in Sherlock and many other exhibitions.
Visit Cumberland House Natural History Museum
At this museum visitors can explore many different types of habitats.
Portsmouth is a very special place for wildlife in Britain and has many different habitats in a very small area.
Displays include dinosaurs, an A to Z of Natural History and a working beehive.
There is also the Butterfly House which is home to a huge array of neo-tropical butterflies, including swallowtails, morphos and owl butterflies.
It is open Tuesday to Sunday, and closed on Monday.
Have an adventure at Canoe Lake
This popular park dates back to 1866 and has a man made lake with pedal boats for hire.
Canoe Lake is also home to a large and very popular children's play area with equipment for all ages and includes a zip wire, sand pit and water area.
There is also a Model Village where everything is on a miniature scale and is known as a swan's nursery as it is one of the mute swans' chosen sites in the Solent.
At times up to 60 juveniles can congregate here for comfort and security during the winter.Γ
Take a trip to Royal Garrison Church
Located in Old Portsmouth this historic church dates back to 1212, so is steeped in history
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CommonCrawl
| 0.030641 | -1.657926 | 0.327162 | 0.297626 |
As the Pythons would chime, "Always Look On the Bright Side of Life."
Foreign Policy editor, David Rothkopf, revisits a lesson he learned from his father that should be welcome in this bleak age. He was 17. His father, a scientist, had been brought by his parents to America before Hitler could take them. Rothkopf had been shaken by a documentary on "nuclear winter" and sought out his dad for advice.
On one beautiful, but for me disturbing, afternoon that summer, I finally found my father outside near the tennis courts at the neighborhood swim club to which we belonged. We sat down and I explained what I had seen. "Hundreds of millions of people will die!" I said, "And it could happen. It could happen any minute. In fact, it probably will happen." The U.S. and Soviet militaries were on a hair-trigger setting. Missiles were waiting in their silos and onboard submarines lurking just off our shores and theirs.
He paused for a minute and then asked with a kind of contrarian perversity that I know he and many other scientists thought was wryly charming, "I see. So, what is it that has you so upset?"
This kind of curveball question had been thrown at me my whole life. But still it made my head spin. "What do you mean, why am I so upset? The whole planet will be devastated. Hundreds of millions will die. Even if you survive, there will be no point in going on living."
He paused for a moment and stared out into the distance. "Well," he said calmly and with a slight trace of a Viennese accent, "you know, a hundred million people β a third of the population of Europe β died during the 14th century of bubonic plague. The result was the Renaissance."
Imagine that you lived during the 14th century. It would be very hard to have much long-term perspective. During the outbreaks of the plague, survival was the only priority. And of course, as bad as it was, the plague was hardly the only concern. The Little Ice Age was beginning.
The Great Schism was dividing the Roman Catholic Church. Mongol rule was ending in the Middle East. The Hundred Years' War had begun. Dynastic upheaval in China ushered in the beginning of the Ming dynasty. The Scots were fighting for independence (some things never change). Much as it is today, Christian Europe and the forces of Islam were in conflict, resulting in the Battle of Kosovo in 1389.
With the advantage of hindsight, we can see that every one of the shocks that rocked the era led not just to substantial progress, but to reordering of the basic way in which society was viewed. In the wake of the human losses of the Black Plague, labor became more valued, and a middle class began to emerge. Trade flows, which may well have brought the plague to Europe from Asia, also led to the exchange of new ideas and materials, to economic and intellectual growth. The nature of work and how we thought of economics began to change.
A weakened Roman Catholic Church began to be challenged by reformers. States began to emerge in forms like those we know today. Ultimately, within a few hundred years of the change, principalities gave way to nation-states, which in turn were locked in a power struggle with the church. The nature of governance had also been transformed.
Universities and scholarship began to take root and spread in new ways. Education for commoners, beginning with literacy, began to spread. Combining the rise of the middle class and the needs of new governments to win support from other powerful members of society, the seeds of more democratic government were planted in places like Europe, themselves predicated on a changing view of the rights of individuals and of states, the role of law, and the nature of communities.
In other words, while the average citizen of the 14th century saw struggle and chaos, changes were afoot that would redefine how people thought of themselves, who they were, what a community was, and of the nature of basic rights, of governance, of work and economics, of war and peace. To understand the future and how it would be different from the past, it would therefore have been essential to consider the questions associated with such changes. To ask in the context of the changing world β and to ask again, amid what was to follow β how does all this change how I view myself, my community, my rights, my government, my
|
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CommonCrawl
| 2.068437 | 0.950622 | 1.81217 | 1.953982 |
job, and the way the world works around me?
As was the case during the 14th century, we too are living in what might be described as the day before the Renaissance. An epochal change is coming, a transformational tsunami is on the horizon, and most of our leaders and many of us have our backs to it. We're looking in the wrong direction. Indeed, many of those in positions of power and their supporters are so actively trying to cling to the past we can almost hear their fingernails clawing at the earth as they try to avoid accepting the inevitable and momentous changes to come.
If we sense that such changes are coming, we have an urgent responsibility to ourselves, our families, and our communities to prepare for them. How do we begin to address these massive shifts in nearly every facet of our lives? How can we begin to prepare for changes that are of a scope and substance that may be greater than any faced for 20 generations, some that may be so great that they force us to reconsider our most fundamental ideas about ourselves and our world? And how can we shift our focus away from the old, comfortable formulations about how societies are organized and operate, what they look like, who should lead them, and what course corrections are essential?
Of course, asking the right questions, and getting the right answers, is easier said than done. We have loads of biases. We expect the world will confirm those biases, and we mishear and misread events around us as a result. We expect the future to be like the past. (After all, we live in a world in which 85 percent of the time the weather tomorrow is the same as the weather today.) We are also harried and so busy reacting to the demands of the moment, much like the average citizen of the 14th century dealing with war and plagues and climatic catastrophes, that pausing to get to the root questions often seems like an impossible luxury, and one we are ill-prepared for. This last point is also important. If we don't understand the technologies or other forces at play in changing our world β be they the burgeoning sciences of the early Renaissance or the neural networks or cyberthreats of today β then how can we possibly understand what is to come?
Furthermore, if those who are supposed to lead us don't understand the changes, they can't ask the right questions either. What's more, they typically have a vested interest in resisting the questions. The status quo got them where they are, and they have a strong interest in preserving it. For example, the predisposition of our political leaders to seek to capitalize on the fears of the moment to advance their self-interested desire to cling to power regularly leads us to keep our eyes on yesterday's headlines rather than on the horizon. Fearmongering is not only exploitative β and, by the way, plays right into the hands of some, like terrorists, who seek to promote fear β it is also a potentially fatal distraction from the bigger risks associated with potential coming changes for which we are ill-prepared.
So, in the end, Hamlet had it wrong. "To be or not to be" is not the question. The question of questions is, "What is the question?" In this respect, history tells us to start with the basics, the foundational questions that we have for too long taken for granted. There are questions like: "Who am I?" "Who rules?" "What is money?" "What is a job?" "What is peace?" and "What is war?"
One lesson is that the more profound the changes, the more basic the questions we should be asking; it is the simplest and most direct questions that cut to the fundamental issues of life, that resist nuance and evasion and rationalization more effectively.
A question like "Who am I?" can lead to questions about how we derive our identity and, in a connected world, how that and our view of communities is likely to change. The answers to those questions can lead us to question whether our old views and systems of governance for communities will work as well in the future, or whether they need to be changed. And they will also raise questions about the role of technology in helping to implement those changes, in creating other kinds of communities, good and bad, driven by our search for identity that might also impact our lives in profound ways.
While there are, as always, a few bright minds out there pioneering new ways of thinking and starting to ask the right questions, it is the responsibility of all of us as citizens to see the questions raised here or in similar discuss
|
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CommonCrawl
| 2.057569 | 1.075209 | 0.345313 | 2.065291 |
10 Horrific Acts Against Women You Won't Believe Still Occur In The 21st Century
by Jessica Eggert
Stocksy
For all those who believe that women are finally reaching equality, think again. As difficult as it is to accept or understand, women are dehumanized in many parts of the world. Sexual, physical and mental abuses are still common 'punishments' for innocent women who just want to achieve equality among men.
In some parts of the world, a woman's right to report allegations of abuse is outweighed by her husband's right to control her. In Asia, 81% of women reported experiencing sexual domestic violence in the past year. The effort to end abuse against women must be brought to the forefront. Here are ten horrific acts of abuse that you won't believe women are experiencing in the 21st century.
A woman's husband chops her fingers off for pursuing a degree.
Hawa Akhtter, a 21-year-old young woman from Bangladesh, had her fingers chopped off by her husband for studying. Her husband warned that if she continued her studies, there would be 'serious consequences.' One day, Hawa's husband told her he had a 'surprise' for her. He blindfolded her, sat her down, took her hand and chopped off all five of her fingers. Akhter says she will still continue her studies.
A woman was sentenced to jail after being raped by a co-worker for having 'extramarital' sex.
Marte Deborah Daley from Norway was on a business trip in Dubai, and she was raped by her co-worker on a trip to the Arab city and sentenced to jail upon reporting it. The police went as far as to ask Daley if she reported the incident as rape simply because the sex was unsatisfying. Under Arab law, rapists are only convicted if there are four adult Muslim male witnesses, or the attacker confesses (yeah, right).
18-year-old Afghan girl gets her nose and ears cut off for running away from her abusive in-laws.
In 2010, a young Afghan girl named Aisha chose to have her picture featured on the cover of TIME magazine, after her ears and nose had been cut off for escaping from in-laws who physically abused her. TIME made sure Aisha was in safe hiding before the issue was published. The Grossman Burn Foundation in California sponsored the young woman's reconstructive surgery.
An American CBS Correspondent, Lara Logan, was sexually attacked and raped in Cairo.
Lara Logan was ripped away from her bodyguard and crew in Cairo, while covering 60 Minutes for CBS, during the time when the Cairo government fell. An estimated 200-300 men were involved in the attack in which they raped and beat Logan, an innocent American woman.
The forced procedure of female circumcision becomes more prevalent.
In parts of the world including Asia, Africa and even Britain, women are forced to have their Labia and Clitoris cut off and their vagina is sewn together, only to allow them to go to the bathroom.
A young Pakistani girl was shot by the Taliban for her women's rights campaign.
Malala Yousafzai was shot in the head by the Taliban on her school bus. The brave young girl still continues to speak up for women's rights. Malala spoke at the UN on her 16th birthday, urging women to pursue an education.
A Yemeni girl runs away from home after being forced into marriage at age 11.
Nada al-Ahdal, a brave young girl from Yemen, ran away from her family when they forced her to get married at age 11. She decided to run away because of her aunt, who had doused herself in gasoline and ignited her own body, killing herself after her arranged married at the age of 13.
A 15-year-old Afghan bride was beaten by her husband and in-laws for refusing to become a prostitute.
Sahar Gul's husband and in-laws burned and beat her, including pulling out her fingernails, after she refused to sell her body for prostitution. Afterwards, the husband and in-laws locked Sahar in a toilet room for six months before being rescued by police.
|
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CommonCrawl
| 0.59579 | 0.327685 | 2.016597 | 0.775117 |
Brexit and Xenophobia vs. Immigration and Innovation
by Cyrus D. Mehta, ABIL Lawyer
In the backlash against globalization, as seen in the vote in favor of Brexit, there is an even more insidious backlash against immigration. The world has prospered because of the expansion of trade and technology, and also due to the free movement of capital and people. Millions of the world's poor people have been lifted from poverty as a result of globalization. In turn, people in richer countries have been able to buy products and services at lower cost. Businesses have also been able to sell goods and services outside beyond national boundaries, thereby becoming more profitable and hiring more people.
Politicians like Donald Trump do not see it this way, who wish to tear up trade deals such as the North American Free Trade Agreement. So does Bernie Sanders, who while speaking with a softer voice, appears to be in harmony with Trump in his critic of globalization and trade deals. While Hillary Clinton is probably in favor of trade deals, she back tracked on the Trans-Pacific Partnership, after being attacked by Sanders during the primaries. It is true that globalization does not always have winners. Those who get displaced need to land on a safety net so that they can re-train and develop new skills. The safety nets, unfortunately, are not keeping up with the enormous changes in technology that increase productivity through innovative technologies, which include rapid strides in robotics and artificial intelligence. During this transition that promises a better future for all in the long run, politicians exploit this shortcoming to lash out against immigrants in their countries and foreign-based workers outside who are paid less, when the true disrupter is technology and innovation.
As Fareed Zakaria so succinctly puts it:
"Manufacturing as a share of all U.S. jobs has been declining for 70 years, as part of a transition experienced by every advanced industrial economy. All other developed countries from Australia to Britain to Germany β which is often seen as a manufacturing powerhouse β have seen similar declines over the past several decades. Even South Korea, which has tried many kinds of protectionism, has experienced a drop in manufacturing as it has become a more advanced economy. This shift is partly a result of free trade, but serious studies show that the much larger cause is technology. One steelworker today makes five times as much steel per hour as he or she did in 1980."
Immigration lawyers know first- hand how free trade and immigration has been beneficial for America. It is due to NAFTA that Canadians and Mexicans can enter the United States on TN visas to work for US employers who seek them out even while the H-1B visa, the main workhorse nonimmigrant visa, has hit the annual numerical cap. Singaporeans and Chileans can enter the United States on H-1B1 visas that ensue from trade deals and so can Australians on an E-3 visa. Nationals of many countries that have treaties with the United States can come here on E-1 and E-2 visas as investors and traders. While the L-1 visa does not ensue from a treaty, it too is premised on the needs of multinational corporations, big and small, in a globalized world. Intra-company transferee managers, executives and specialized workers can work for a US branch, subsidiary, parent or affiliate of a foreign company on L-1 visas. Despite there not being H-1B visas, the fact that other visas are still available, allow US companies to remain globally competitive by tapping into skilled and professional foreign workers. If it were not for these visas, the entry of skilled workers into America would be at a standstill.
We need to embrace immigrants, and view them as an asset, rather than as people who steal jobs and work cheaply. Immigration not only provides a complimentary workforce, but also generates innovation that will create the next generation of jobs that require new skills. If we have a robust and welcoming immigration system that would not shackle the worker to one employer, but would allow mobility and a quick pathway to permanent residency, then there would be no suppression of wages. Everyone would be on a level playing field, and market forces would ensure that wages remain competitive. Indeed, by encouraging more movement of people to America and other richer countries,
|
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CommonCrawl
| 1.000632 | 0.649536 | 0.983701 | 1.144461 |