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2,700 |
A new measure of growth for countable-dimensional algebras
|
math.RA
|
A new dimension function on countable-dimensional algebras (over a field) is
described. Its dimension values for finitely generated algebras exactly fill
the unit interval $[0,1]$. Since the free algebra on two generators turns out
to have dimension 0 (although conceivably some Noetherian algebras might have
positive dimension!), this dimension function promises to distinguish among
algebras of infinite GK-dimension.
|
math
|
2,701 |
Orthomodularity in infinite dimensions; a theorem of M. Solèr
|
math.RA
|
Maria Pia Sol\`er has recently proved that an orthomodular form that has an
infinite orthonormal sequence is real, complex, or quaternionic Hilbert space.
This paper provides an exposition of her result, and describes its consequences
for Baer $\ast$-rings, infinite-dimensional projective geometries, orthomodular
lattices, and Mackey's quantum logic.
|
math
|
2,702 |
A number of countable models of a countable supersimple theory
|
math.RA
|
In this paper, we prove the number of countable models of a countable
supersimple theory is either 1 or infinite. This result is an extension of
Lachlan's theorem on a superstable theory.
|
math
|
2,703 |
The relationship between two commutators
|
math.RA
|
We clarify the relationship between the linear commutator and the ordinary
commutator by showing that in any variety satisfying a nontrivial idempotent
Mal'cev condition the linear commutator is definable in terms of the
centralizer relation. We derive from this that abelian algebras are
quasi-affine in such varieties. We refine this by showing that if A is an
abelian algebra and V(A) satifies an idempotent Mal'cev condition which fails
to hold in the variety of semilattices, then A is affine.
|
math
|
2,704 |
Projectivity and isomorphisms of strictly simple algebras
|
math.RA
|
We describe a sufficient condition for the localization functor to be a
categorical equivalence. Using this result we explain how to simplify the test
for projectivity. This leads to a description of the strictly simple algebras
which are projective in the variety they generate. A byproduct of our efforts
is the result that if A and B are strictly simple and generate the same
variety, then A=B or else both are strongly abelian.
|
math
|
2,705 |
Almost orthogonal submatrices of an orthogonal matrix
|
math.RA
|
Let $A$ be an $n \times M$ matrix whose rows are orthonormal. Let $A_I$ be a
submatrix of $A$ whose column indexes belong to the set $I$. Given $\epsilon
>0$ we estimate the smallest cardinality of the set $I$, such that the operator
$A_I$ is an $\epsilon$-isometry.
|
math
|
2,706 |
Self-rectangulating varieties of type 5
|
math.RA
|
We show that a locally finite variety which omits abelian types is
self-regulating if and only if it has a compatible semilattice term operation.
Such varieties must have a type-set {5}. These varieties are residually small
and, when they are finitely generated, they have definable principal
congruences. We show that idempotent varieties with a compatible semilattice
term operation have the congruence extension property.
|
math
|
2,707 |
Tensor product representations for orthosymplectic Lie superalgebras
|
math.RA
|
We derive a general result about commuting actions on certain objects in
braided rigid monoidal categories. This enables us to define an action of the
Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the
action of the orthosymplectic Lie superalgebra $\spo(V)$ and the
orthosymplectic Lie color algebra $\spo(V,\beta)$. We use the Brauer algebra
action to compute maximal vectors in $V^{\otimes k}$ and to decompose
$V^{\otimes k}$ into a direct sum of submodules $T^\lambda$. We compute the
characters of the modules $T^\lambda$, give a combinatorial description of
these characters in terms of tableaux, and model the decomposition of
$V^{\otimes k}$ into the submodules $T^\lambda$ with a Robinson-Schensted-Knuth
type insertion scheme.
|
math
|
2,708 |
A note on Lascar strong types in simple theories
|
math.RA
|
Let T be a countable, small simple theory. In this paper, we prove for such
T, the notion of Lascar Strong type coincides with the notion of a strong
type,over an arbitrary set.
|
math
|
2,709 |
Modularity prevents tails
|
math.RA
|
We establish a direct correspondence between two congruence poroperties for
finite algebras. The first property is that minimal sets of type i omit tails.
The second property is that congruence lattices omit pentagons of type i.
|
math
|
2,710 |
Type 4 is not computable
|
math.RA
|
We extend a recent result of McKenzie, and show that it is an undecidable
problem to determine if 4 appears in the typeset of a finitely generated,
locally finite variety.
|
math
|
2,711 |
Closures in $\aleph_0$-categorical bilinear maps
|
math.RA
|
Alternating bilinear maps with few relations allow to define a combinatorial
closure similarly as in [2]. For the $\aleph_0$-categorical case we show that
this closure is part of the algebraic closure.
|
math
|
2,712 |
Rigid analytic flatificators
|
math.RA
|
Let K be an algebraically closed field endowed with a complete
non-archimedean norm. Let f:Y -> X be a map of K-affinoid varieties. We prove
that for each point x in X, either f is flat at x, or there exists, at least
locally around x, a maximal locally closed analytic subvariety Z in X
containing x, such that the base change f^{-1}(Z) -> Z is flat at x, and,
moreover, g^{-1}(Z) has again this property in any point of the fibre of x
after base change over an arbitrary map g:X' -> X of affinoid varieties. If we
take the local blowing up \pi:X-tilde -> X with this centre Z, then the fibre
with respect to the strict transform f-tilde of f under \pi, of any point of
X-tilde lying above x, has grown strictly smaller. Among the corollaries to
these results we quote, that flatness in rigid analytic geometry is local in
the source; that flatness over a reduced quasi-compact rigid analytic variety
can be tested by surjective families; that an inclusion of affinoid domains is
flat in a point, if it is unramified in that point.
|
math
|
2,713 |
A theorem on spherically complete valued abelian groups
|
math.RA
|
We give a criterion for a group homomorphism on a valued abelian group to be
surjective and to preserve spherical completeness. We apply this to give a
criterion for the existence of integration on a valued differential field.
Further, we give a criterion for a sum of spherically complete subgroups of a
valued abelian group to be spherically complete. This in turn can be used to
determine elementary properties of power series fields in positive
characteristic.
|
math
|
2,714 |
Z_8 is not dualizable
|
math.RA
|
In this paper we show that Z_8 does not admit a natural duality. In fact, we
show that 2Z_8 = {2, 4, 6, 8 | +,.} is not dualizable, and this will imply that
the original ring is not dualizable, either. As a corollary we show that
Sindi's conjecture does not hold. Our technique will be similar to one due to
Quackenbush and Szab\'o, where non-dualizability is proved for the quaternion
group.
|
math
|
2,715 |
The Pfaffian closure on an o-minimal structure
|
math.RA
|
Every o-minimal expansion R-tilde of the real field has an o-minimal
expansion P(R-tilde) in which the solutions to Pfaffian equations with
definable C^1 coefficients are definable.
|
math
|
2,716 |
Complexity problems associated with matrix rings, matrix semigroups and Rees matrix semigroups
|
math.RA
|
Complexity problems associated with finite rings and finite semigroups,
particularly semigroups of matrices over a field and the Rees matrix
semigroups, are examined. Let M_nF be the ring of n x n matrices over the
finite field F and let T_nF be the multiplicative semigroup of n x n matrices
over the finite field F. It is proved that for any finite field F and positive
integer n >= 2, the polynomial equivalence problem for the T_n F is
co-NPcomplete, thus POL-EQ_\Sigma(M_n F) (POL-EQ_\Sigma is a polynomial
equivalence problem for which polynomials are presented as sums of monomials)
is also co-NP-complete thereby resolving a problem of J. Lawrence and R.
Willard and completing the description of POL-EQ_\Sigma for the finite simple
rings. In connection with our results on rings, we exhibit a large class of
combinatorial Rees matrix semigroups whose polynomial equivalence problem is
co-NP-complete. On the other hand, if S is a combinatorial Rees matrix
semigroup with a totally balanced structure matrix M, then we prove that the
polynomial equivalence problem for S is in P. Fully determining the complexity
of the polynomial equivalence problem for combinatorial Rees matrix semigroups
may be a difficult problem. We describe a connection between the polynomial
equivalence problem for combinatorial Rees matrix semigroups and the retraction
problem RET for bipartite graphs, a problem which computer scientists suspect
may not admit a dichotomy into P and NP-complete problems (assuming P is not
equal to NP).
|
math
|
2,717 |
Primitive representations of finite semigroups I
|
math.RA
|
Primitive representations of finite groups as well as primitive finite groups
were classified in the O'Nan-Scott Theorem. In this paper we classify faithful
finite primitive semigroup representations. To each finite primitive
representation, we associate an invariant, a finite dimensional matirix with
entries in a primitive finite groups representation. To a large extent, this
matrix determines the representation. In later papers in this series the
invariant is further explored. Our primitivity resoullts rely on two main
ideas, one of which is a small but important part of the theory of tame
algebras developed by R. McKenzie, while the other is our adaptation of Rees
matrix theory to the representation theory of finite semigroups. Using the
latter idea we are able to provide a description of all representations of a
regular T class of a finite semigroup.
|
math
|
2,718 |
A finite basis theorem for residually finite, congruence meet-semidistributive varieties
|
math.RA
|
We derive a Mal'cev condition for congruence meet-semidistributivity and then
use it to prove two theorems. Theorem A: if a variety in a finite language is
congruence meet-semidistributive and residually less than some finite cardinal,
then it is finitely based. Theorem B: there is an algorithm which, given m<w
and a finite algebra in a finite language, determines whether the variety
generated by the algebra is congruence meet-semidistributive and residually
less then m.
|
math
|
2,719 |
Associative algebras satisfying a semigroup identity
|
math.RA
|
Denote by (R,.) the multiplicative semigroup of an associative algebra R over
an infinite field, and let (R,*) represent R when viewed as a semigroup via the
circle operation x*y=x+y+xy. In this paper we characterize the existence of an
identity in these semigroups in terms of the Lie structure of R. Namely, we
prove that the following conditions on R are equivalent: the semigroup (R,*)
satisfies an identity; the semigroup (R,.) satisfies a reduced identity; and,
the associated Lie algebra of R satisfies the Engel condition. When R is
finitely generated these conditions are each equivalent to R being upper Lie
nilpotent.
|
math
|
2,720 |
Bounds on norms of compound matrices and on products of eigenvalues
|
math.RA
|
An upper bound on operator norms of compound matrices is presented, and
special cases that involve the $\ell_1$, $\ell_2$ and $\ell_\infty$ norms are
investigated. The results are then used to obtain bounds on products of the
largest or smallest eigenvalues of a matrix.
|
math
|
2,721 |
On matrices for which norm bounds are attained
|
math.RA
|
Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$
columns by the H\"older $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or
$C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio
$\|A\|_{r,s}/\|A\|_{p,q}$. In this paper we study the classes of matrices for
which the upper bound is attained. We shall show that for fixed $A$, attainment
of the bound depends only on the signs of $r-p$ and $s-q$. Various criteria
depending on these signs are obtained. For the special case $p=q=2$, the set of
all matrices for which the bound is attained is generated by means of singular
value decompositions.
|
math
|
2,722 |
On the $n$-ary algebras, semigroups and their universal covers
|
math.RA
|
For any $n$-ary associative algebra we construct a $\Z_{n-1}$ graded algebra,
which is a universal object containing the $n$-ary algebra as a subspace of
elements of degree 1. Similar construction is carried out for semigroups.
|
math
|
2,723 |
Modules Whose Small Submodules Have Krull Dimension
|
math.RA
|
The main aim of this paper is to show that an AB5*-module whose small
submodules have Krull dimension has a radical having Krull dimension. The proof
uses the notion of dual Goldie dimension.
|
math
|
2,724 |
On Semilocal Modules and Rings
|
math.RA
|
It is well-known that a ring R is semiperfect if and only if R as a left (or
as a right) R-module is a supplemented module. Considering weak supplements
instead of supplements we show that weakly supplemented modules M are semilocal
(i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R
as a left (or as a right) R-module is weakly supplemented. In this context the
notion of finite hollow dimension (or finite dual Goldie dimension) of modules
is of interest and yields a natural interpretation of the Camps-Dicks
characterization of semilocal rings. Finitely generated modules are weakly
supplemented if and only if they have finite hollow dimension (or are
semilocal).
|
math
|
2,725 |
Hermitian Positive Semidefinite Matrices Whose Entries Are 0 Or 1 in Modulus
|
math.RA
|
We show that a matrix is a Hermitian positive semidefinite matrix whose
nonzero entries have modulus 1 if and only if it similar to a direct sum of all
$1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular,
the only such nonsingular matrix is the identity matrix and the only such
irreducible matrix is similar to an all 1's matrix by means of a unitary
diagonal similarity. Our results extend earlier results of Jain and Snyder for
the case in which the nonzero entries (actually) equal 1. Our methods of proof,
which rely on the so called principal submatrix rank property, differ from the
approach used by Jain and Snyder.
|
math
|
2,726 |
The Octonionic Eigenvalue Problem
|
math.RA
|
We discuss the eigenvalue problem for 2x2 and 3x3 octonionic Hermitian
matrices. In both cases, we give the general solution for real eigenvalues, and
we show there are also solutions with non-real eigenvalues.
|
math
|
2,727 |
Principal pivot transforms: properties and applications
|
math.RA
|
The principal pivot transform (PPT) of a matrix A partitioned relative to an
invertible leading principal submatrix is a matrix B such that
A [x_1^T x_2^T]^T = [y_1^T y_2^T]^T
if and only if
B [y_1^T x_2^T]^T = [x_1^T y_2^T]^T,
where all vectors are partitioned conformally to A. The purpose of this paper
is to survey the properties and manifestations of PPTs relative to arbitrary
principal submatrices, make some new observations, present and possibly
motivate further applications of PPTs in matrix theory. We pay special
attention to PPTs of matrices whose principal minors are positive.
|
math
|
2,728 |
Finding Octonionic Eigenvectors Using Mathematica
|
math.RA
|
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some
surprises, which we have reported elsewhere. In particular, the eigenvalues
need not be real, there are 6 rather than 3 real eigenvalues, and the
corresponding eigenvectors are not orthogonal in the usual sense. The
nonassociativity of the octonions makes computations tricky, and all of these
results were first obtained via brute force (but exact) Mathematica
computations. Some of them, such as the computation of real eigenvalues, have
subsequently been implemented more elegantly; others have not. We describe here
the use of Mathematica in analyzing this problem, and in particular its use in
proving a generalized orthogonality property for which no other proof is known.
|
math
|
2,729 |
Polycyclic-by-finite group algebras are catenary
|
math.RA
|
We show that group algebras kG of polycyclic-by-finite groups G, where k is a
field, are catenary: Given prime ideals P and P' of kG, with P contained in P',
all saturated chains of primes between P and P' have the same length.
|
math
|
2,730 |
On the Splitting of the Dual Goldie Torsion Theory
|
math.RA
|
The splitting of the Goldie (or singular) torsion theory has been extensively
studied. Here we determine an appropriate dual Goldie torsion theory, discuss
its splitting and answer in the negative a question proposed by Ozcan and
Harmanci as to whether the splitting of the dual Goldie torsion theory implies
the ring to be quasi-Frobenius.
|
math
|
2,731 |
Supersimple fields and division rings
|
math.RA
|
It is proved that any supersimple field has trivial Brauer group, and more
generally that any supersimple division ring is commutative. As prerequisites
we prove several results about generic types in groups and fields whose theory
is simple.
|
math
|
2,732 |
Lascar and Morley ranks differ in differentially closed fields
|
math.RA
|
We note here, in answer to a question of Poizat, that the Morley and Lascar
ranks need not coincide in differentially closed fields. We approach this
through the (perhaps) more fundamental issue of the variation of Morley rank in
families.
|
math
|
2,733 |
On finite homomorphic images of the multiplicative group of a division algebra
|
math.RA
|
This paper, together with a forthcoming paper by the author and Seitz, proves
the Margulis-Platonov conjecture concerning the normal subgroup structure of
algebraic groups over number fields, in the case of inner forms of anisotropic
groups of type $A_n$.
|
math
|
2,734 |
Growth and Relations in Graded Rings
|
math.RA
|
Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal
generated by a set $\alpha$ of homogeneous elements, and B = A/I. In this note,
some inequalities between Hilbert series of algebras $A,B$ and the number of
elements of the set $\alpha$ are announced. As in the Golod--Shafarevich
inequality as in our case the equality in every estimate is exact iff the set
$\alpha$ is strongly free: so we obtain some new characterizations of such
sets. As a consequence it is proved that over a field of zero characteristic
for the class of finitely defined graded algebras there is no algorithm to
answer the following question: for an algebra $A$ and a rational number $R$, is
the convergence radius of the Hilbert series of $A$ equal to $R$?
|
math
|
2,735 |
One-sided Noncommutative Groebner Bases with Applications to Green's Relations
|
math.RA
|
Standard noncommutative Gr\"obner basis procedures are used for computing
ideals of free noncommutative polynomial rings over fields. This paper
describes Gr\"obner basis procedures for one-sided ideals in finitely presented
noncommutative algebras over fields. The polynomials defining a $K$-algebra $A$
as a quotient of a free $K$-algebra are combined with the polynomials defining
a one-sided ideal $I$ of $A$, by using a tagging notation. Standard
noncommutative Gr\"obner basis techniques can then be applied to the mixed set
of polynomials, thus calculating $A/I$ whilst working in a free structure,
avoiding the complication of computing in $A$. The paper concludes by showing
how the results can be applied to completable presentations of semigroups and
so enable calculations of Green's relations.
|
math
|
2,736 |
Enumeration of Finite Rings with Jacobson Radical of Cube Zero
|
math.RA
|
In [1], finite associative rings wih identity and such that the set of all
zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and
square non-zero, were constructed for all the characteristics. These rings are
completely primary and we call them rings with property(T). In this paper, we
associate with each ring with property(T) and characteristic p, invariants
(integers) and determine (in certain cases) the number of isomorphism classes
of these rings with given invariants.
|
math
|
2,737 |
The graded version of Goldie's Theorem
|
math.RA
|
The analogue of Goldie's Theorem for prime rings is proved for rings graded
by abelian groups, eliminating unnecessary additional hypotheses used in
earlier versions.
|
math
|
2,738 |
Generalized Selective Modal Analysis
|
math.RA
|
A new approach which generalizes the Selective Modal Analyis (SMA) and
algorithms based upon it for solving the generalized eigenvalue problem is
described. This approach allows for the systematic consideration of physical
properties of the system under study. Two small application cases demonstrate
the capabilities of the proposed approach.
|
math
|
2,739 |
Graded Lie Algebras of Maximal Class II
|
math.RA
|
We describe the isomorphism classes of infinite-dimensional graded Lie
algebras of maximal class, generated by elements of weight one, over fields of
odd characteristic.
|
math
|
2,740 |
Graded Lie algebras of maximal class IV
|
math.RA
|
We describe the isomorphism classes of certain infinite-dimensional graded
Lie algebras of maximal class, generated by an element of weight one and an
element of weight two, over fields of odd characteristic.
|
math
|
2,741 |
Nahm Algebras
|
math.RA
|
Given a Lie algebra $\frak{g}$, the \emph{Nahm algebra} of $\frak{g}$ is the
vector space $\frak{g}\times \frak{g}\times \frak{g}$ with the natural
commutative, nonassociative algebra structure associated with the Nahm
equations $\dot{x} = [y,z]$, $\dot{y} = [z,x]$, $\dot{z} = [x,y]$. Motivated by
potential application to the study of these equations, we herein initiate the
study of Nahm algebras.
|
math
|
2,742 |
Hecke Algebras, SVD, and Other Computational Examples with {\sc CLIFFORD}
|
math.RA
|
{\sc CLIFFORD} is a Maple package for computations in Clifford algebras $\cl
(B)$ of an arbitrary symbolic or numeric bilinear form B. In particular, B may
have a non-trivial antisymmetric part. It is well known that the symmetric part
g of B determines a unique (up to an isomorphism) Clifford structure on
$\cl(B)$ while the antisymmetric part of B changes the multilinear structure of
$\cl(B).$ As an example, we verify Helmstetter's formula which relates Clifford
product in $\cl(g)$ to the Clifford product in $\cl(B).$ Experimentation with
Clifford algebras $\cl(B)$ of a general form~B is highly desirable for physical
reasons and can be easily done with {\sc CLIFFORD}. One such application
includes a derivation of a representation of Hecke algebras in ideals generated
by q-Young operators. Any element (multivector) of $\cl(B)$ is represented in
Maple as a multivariate Clifford polynomial in the Grassmann basis monomials
although other bases, such as the Clifford basis, may also be used. Using the
well-known isomorphism between simple Clifford algebras $\cl(Q)$ of a quadratic
form Q and matrix algebras through a faithful spinor representation, one can
translate standard matrix algebra problems into the Clifford algebra language.
We show how the Singular Value Decomposition of a matrix can be performed in a
Clifford algebra. Clifford algebras of a degenerate quadratic form provide a
convenient tool with which to study groups of rigid motions in robotics. With
the help from {\sc CLIFFORD} we can actually describe all elements of $\Pin(3)$
and $\Spin(3).$ Rotations in $\BR^3$ can then be generated by unit quaternions
realized as even elements in $\cl^{+}_{0,3}.$ Throughout this work all symbolic
computations are performed with {\sc CLIFFORD} and its extensions.
|
math
|
2,743 |
FP-injective and weakly quasi-Frobenius rings
|
math.RA
|
The classes of FP-injective and weakly quasi-Frobenius rings are
investigated. The properties for both classes of rings are closely linked with
embedding of finitely presented modules in fp-flat and free modules
respectively. Using these properties, we describe the classes of coherent CF
and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective
(resp. weakly quasi-Frobenius) if and only if the ring R is FP-injective (resp.
weakly quasi-Frobenius) and the group G is locally finite.
|
math
|
2,744 |
Index of Hadamard multiplication by positive matrices II
|
math.RA
|
Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal
index of A: $I(A) = \max \{\lambda \ge 0 : A\circ B \ge \lambda B$ for all
$0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} =
A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the
N-index of A: $I(N,A) = \min\{N(A\circ B) : B\ge 0$ and $N(B) = 1 \}$. If A has
nonnegative entries, then $I(A) = I(\| \cdot \|_{sp}, A)$ if and only if there
exists a vector u with nonnegative entries such that $Au = (1, >..., 1)^T$. We
also show that $I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ
A)^{1/2}$. We give formulae for I(N, A), for an arbitrary unitary invariant
norm N, when A is a diagonal matrix or a rank 1 matrix. As an application we
find, for a bounded invertible selfadjoint operator S on a Hilbert space, the
best constant M(S) such that $\|STS + S^{-1} T S^{-1} \| \ge M(S) \|T\| $ for
all $0 \le T$.
|
math
|
2,745 |
Bass's Work in Ring Theory and Projective Modules
|
math.RA
|
The early papers of Hyman Bass in the late 50s and the early 60s leading up
to his pioneering work in algebraic K-theory have played an important and very
special role in ring theory and the theory of projective (and injective)
modules. In this article, we give a general survey of Bass's fundamental
contributions in this early period of his work, and explain how much this work
has influenced and shaped the thinking of subsequent researchers in the area.
|
math
|
2,746 |
Interpolation in ortholattices
|
math.RA
|
If L is a complete ortholattice, f any partial function from L^n to L, then
there is a complete ortholattice L* containing L as a subortholattice, and an
ortholattice polynomial with coefficients in L* which represents f on L^n.
Iterating this construction long enough yields a complete ortholattice in
which every function can be interpolated by a polynomial on any set of small
enough cardinality.
|
math
|
2,747 |
On the degrees of irreducible representations of Hopf algebras
|
math.RA
|
Let H denote a semisimple Hopf algebra over an algebraically closed field k
of characteristic 0. We show that the degree of any irreducible representation
of H whose character belongs to the center of H^* must divide the dimension of
H .
|
math
|
2,748 |
Matrix Representations of Octonions and Their Applications
|
math.RA
|
As is well-known, the real quaternion division algebra $ {\cal H}$ is
algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division
octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix
algebras over the real number field ${\cal R}$, because ${\cal O}$ is a
non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an
extension of ${\cal H}$ by the Cayley-Dickson process and is also
finite-dimensional, some pseudo real matrix representations of octonions can
still be introduced through real matrix representations of quaternions. In this
paper we give a complete investigation to real matrix representations of
octonions, and consider their various applications to octonions as well as
matrices of octonions.
|
math
|
2,749 |
Rank Equalities Related to Generalized Inverses of Matrices and Their Applications
|
math.RA
|
This paper is divided into two parts. In the first part, we develop a general
method for expressing ranks of matrix expressions that involve Moore-Penrose
inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose
inverses of matrices. Through this method we establish a variety of valuable
rank equalities related to generalized inverses of matrices mentioned above.
Using them, we characterize many matrix equalities in the theory of generalized
inverses of matrices and their applications. In the second part, we consider
maximal and minimal possible ranks of matrix expressions that involve variant
matrices, the fundamental work is concerning extreme ranks of the two linear
matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications,
we present a wide range of their consequences and applications in matrix
theory.
|
math
|
2,750 |
Matrix Theory over the Complex Quaternion Algebra
|
math.RA
|
We present in this paper some fundamental tools for developing matrix
analysis over the complex quaternion algebra. As applications, we consider
generalized inverses, eigenvalues and eigenvectors, similarity, determinants of
complex quaternion matrices, and so on.
|
math
|
2,751 |
Finite Groups Embeddable in Division Rings
|
math.RA
|
Finite groups that are embeddable in the multiplicative groups of division
rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has
characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic
groups, according to a theorem of I. N. Herstein. Thus, the only interesting
case is when $K$ has characteristic 0; that is, when $K\supseteq {\Bbb Q}$.
Herstein conjectured that odd-order subgroups of division rings $K$ were
cyclic, and he proved this to be the case when $K$ is the division ring of the
real quaternions. Herstein's conjecture was settled negatively by Amitsur. As
part of his complete classification of finite groups in division rings, Amitsur
showed that the smallest noncyclic odd-order group that can be embedded in a
division ring is one of order 63 (and this group is unique).
Amitsur's paper is daunting to read as it is long and technically
complicated. In lecturing to a graduate class on division rings, I tried to
find a simple reason for the ``first exceptional odd order'' 63 (to Herstein's
conjecture). After some work, I did come up with a reason that was simple
enough to be explained to my class, without having to go through any part of
Amitsur's paper. Furthermore, the method I used led easily to the second
exceptional odd order, 117 (which was not mentioned in Amitsur's paper). Since
this line of reasoning did not seem to have appeared in the literature before,
I record it in this short note. To better motivate the results discussed here,
I have also included a quick exposition on the beginning part of the theory of
finite subgroups of division rings.
|
math
|
2,752 |
Jonssons's theorem in non-modular varieties
|
math.RA
|
A version of Jonsson's theorem, as previously generalized, holds in
non-modular varieties.
|
math
|
2,753 |
Regular coverings in filter and ideal lattices
|
math.RA
|
The Dedekind-Birkhoff theorem for finite-height modular lattices has
previously been generalized to complete modular lattices using the theory of
regular coverings. In this paper, we investigate regular coverings in lattices
of filters and lattices of ideals, and the regularization strategy--embedding
the lattice into its lattice of filters or lattice of ideals, thereby possibly
converting a covering which is not regular into a covering which is regular.
One application of the theory is a generalization of the notion of chief
factors, and of the Jordan-Holder Theorem, to cases where the modular lattice
in question is of infinite height. Another application is a formalization of
the notion of the steps in the proof of a theorem.
|
math
|
2,754 |
Abelian extensions of algebras in congruence-modular varieties
|
math.RA
|
We define abelian extensions of algebras in congruence-modular varieties. The
theory is sufficiently general that it includes, in a natural way, extensions
of R-modules for a ring R. We also define a cohomology theory, which we call
clone cohomology, such that the cohomology group in dimension one is the group
of equivalence classes of extensions.
|
math
|
2,755 |
Algebras with a compatible uniformity
|
math.RA
|
Given a variety of algebras V, we study categories of algebras in V with a
compatible structure of uniform space. The lattice of compatible uniformities
of an algebra, Unif A, can be considered a generalization of the lattice of
congruences Con A. Mal'cev properties of V influence the structure of Unif A,
much as they do that of Con A. The category V[CHUnif] of complete, Hausdorff
uniform algebras in the variety V is particularly interesting; it has a natural
factorization system extending the usual (onto, one-one) factorization system
of V.
|
math
|
2,756 |
Octonionic Hermitian Matrices with Non-Real Eigenvalues
|
math.RA
|
We extend previous work on the eigenvalue problem for Hermitian octonionic
matrices by discussing the case where the eigenvalues are not real, giving a
complete treatment of the 2x2 case, and summarizing some prelimenary results
for the 3x3 case.
|
math
|
2,757 |
Algebras without noetherian filtrations
|
math.RA
|
We provide examples of finitely generated noetherian PI algebras for which
there is no finite dimensional filtration with a noetherian associated graded
ring; thus we answer negatively a question raised by M. Lorenz.
|
math
|
2,758 |
Nonfiliform characteristically nilpotent Lie algebras
|
math.RA
|
We construct large families of characteristically nilpotent Lie algebras by
considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and
which arises as a central extension fo the filiform Lie algebra L_{n}. By
studying the graded cohomology spaces we obtain that the sill algebras are
isomorphic to the nilradicals of solvable, complete Lie algebra laws. For
extremal cocycles these laws are also rigid. Considering supplementary cocycles
we construcy, for dimensions n>8, nonfiliform characteristically nilpotent Lie
algebras and show that for certain deformations these are compatible with
central extensions.
|
math
|
2,759 |
Computing homomorphisms between holonomic D-modules
|
math.RA
|
Let K be a subfield of the complex numbers, and let D be the Weyl algebra of
K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic
left D-modules we present an algorithm that computes explicit generators for
the finite dimensional vector space hom_D(M,N). This enables us to answer
algorithmically whether two given holonomic modules are isomorphic. More
generally, our algorithm can be used to get explicit generators for
ext^i_D(M,N) for any i.
|
math
|
2,760 |
On k-abelian, p-filiform Lie algebras
|
math.RA
|
We classify the (n-5)-filiform Lie algebras which have the additional
property of a non-abelian derived subalgebra. We show that this property is
strongly related with the structure of the Lie algebra of derivations;
explicitely we show that if a (n-5)-filiform algebra is characteristically
nilpotent, then it must be 2-abelian. We also give applications of k-abelian
Lie algebras to the construction of solvable rigis algebras, as well as to the
theory of nilalgebras of parabolic subalgebras in the example of the
exceptional simple model E_{6}.
|
math
|
2,761 |
Wedderburn Polynomials over Division Rings
|
math.RA
|
A Wedderburn polynomial over a division ring K is a minimal polynomial of an
algebraic subset of K. Special cases of such polynomials include, for instance,
the minimal polynomials (over the center F=Z(K)) of elements of K that are
algebraic over F. In this note, we give a survey on some of our ongoing work on
the structure theory of Wedderburn polynomials. Throughout the note, we work in
the general setting of an Ore skew polynomial ring K[t,S,D].
|
math
|
2,762 |
Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras
|
math.RA
|
In this article we prove that for a basic classical Lie superalgebra the
annihilator of a strongly typical Verma module is a centrally generated ideal.
For a basic classical Lie superalgebra of type I we prove that the localization
of the enveloping algebra by a certain central element is free over its centre.
|
math
|
2,763 |
On certain families of naturally graded Lie algebras
|
math.RA
|
In this work large families of naturally graded nilpotent Lie algebras in
arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying
the centralizer property, are given. This condtion constitutes a
generalization, for a nilpotent Lie agebra, of the structural properties
charactrizing the Lie algebra $Q_{n}$. By considering certain cohomological
classes of the space $H^{2}(\frak{g},\mathbb{C})$, it is shown that, with few
exceptions, the isomorphism classses of these algebras are given by central
extensions of $Q_{n}$ by $\mathbb{C}^{p}$ which preserve the nilindex and the
natural graduation.
|
math
|
2,764 |
On weight graphs for nilpotent Lie algebras I
|
math.RA
|
We introduce the concept of weight graph for the weight system $P\frak{g}(T)$
of a finite dimensional nilpotent Lie algebra $\frak{g}$ and analyze the
necessary conditions for a $(p,q)$-graph to be a weight graph for some
$\frak{g}$.
|
math
|
2,765 |
Some Properties of 3x3 Octonionic Hermitian Matrices with Non-Real Eigenvalues
|
math.RA
|
We discuss our preliminary attempts to extend previous work on 2x2 Hermitian
octonionic matrices with non-real eigenvalues to the 3x3 case.
|
math
|
2,766 |
Duality and Rational Modules in Hopf Algebras over Commutative Rings
|
math.RA
|
Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and
$A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules
and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can
also strengthen the Blattner-Montgomery duality theorem. Finally, we give
sufficient conditions to get the purity of $A^\circ$ in $R^A$.
|
math
|
2,767 |
Characteristically nilpotent Lie algebras : a survey
|
math.RA
|
We review the known results about characteristically nilpotent complex Lie
algebras, as well as we comment recent developements in the theory.
|
math
|
2,768 |
On the determination of 2-step solvable Lie algebra from its weight graph
|
math.RA
|
By using the concept of weight graph associated to certain nilpotent Lie
algebras $\frak{g}$, we find necessary and sufficient conditions for a
semidirect product $\frak{g}\oplus T_{i}$, where $T_{i}<T$ is a subalgebra of a
maximal torus of derivations $T$ of $\frak{g}$ which induces a decomposition of
$\frak{g}$ into one dimensional weight spaces, to be 2-step solvable. In
particular we show that the semidirect product of such a Lie algebra with its
torus of derivations cannot be itself 2-step solvable.
|
math
|
2,769 |
Integration on Lie supergroups. A Hopf superalgebra approach
|
math.RA
|
For a large class of finite-dimensional Lie superalgebras (including the
classical simple ones) a Lie supergroup associated to the algebra is defined by
fixing the Hopf superalgebra of functions on the supergroup. Then it is shown
that on this Hopf superalgebra there exists a non-zero left integral. According
to a recent work by the authors, this integral is unique up to scalar
multiples.
|
math
|
2,770 |
On a generic inverse differential Galois problem for GL_n
|
math.RA
|
\newcommand{\GLn}{\operatorname{GL}_n} \newcommand{\GL}{\GLn(C)}
Let $F$ be a differential field with algebraically closed field of constants
$C$. We prove that $F< Y_{ij}>(X_{ij})\supset F< Y_{ij}>$ is a generic
Picard-Vessiot extension of $F$ for $\GL$. If $E\supset F$ is any
Picard-Vessiot extension with differential Galois group $\GL$ then $E\cong
F(X_{ij})$ as $F$- and $\GL$-modules and there are $f_{ij}\in F$ such that $F<
Y_{ij}>(X_{ij})\supset F< Y_{ij}>$ specializes to $E\supset F$ via $
Y_{ij}\mapsto f_{ij}$. The $[f_{ij}]\in M_n(F)$ for which the image of the map
$ Y_{ij}\mapsto f_{ij}$ is a Picard-Vessiot extension of $F$ with group $\GL$
can be characterized as those $[f_{ij}]\in M_n(F)$ for which the wronskians of
the monomials in $F< Y_{ij}>(X_{ij})$ of degree less than or equal to $k$ all
map to non-zero elements under $ Y_{ij}\mapstof_{ij}$.
|
math
|
2,771 |
Classification of (n-5)-filiform Lie algebras
|
math.RA
|
In this paper we consider the problem of classifying the $(n-5)$-filiform Lie
algebras. This is the first index for which infinite parametrized families
appear, as can be seen in dimension $7.$ Moreover we obtain large families of
characteristic nilpotent Lie algebras with nilpotence index 5 and show that at
least for dimension 10 there is a characteristic nilpotent Lie algebra with
nilpotence index 4 which is the algebra of derivations of a nilpotent Lie
algebra.
|
math
|
2,772 |
Exact interval solutions to the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra
|
math.RA
|
In this note we construct a solution of a matrix interval linear equation of
the form X=AX+B (the discrete stationary Bellman equation) over partially
ordered semirings, including the semiring of nonnegative real numbers and all
idempotent semirings. We discuss also the computational complexity of problems
in interval idempotent linear algebra. In the traditional Interval Analysis
problems of this kind are generally NP-hard. In the note we consider matrix
equations over positive semirings; in this case the computational complexity of
the problem is polynomial.
Idempotent and other positive semirings arise naturally in optimization
problems. Many of these problems turn out to be linear over appropriate
idempotent semirings. In this case, the system of equations X=AX+B appears to
be a natural analog of a usual linear system in the traditional linear algebra
over fields. B. A. Carre showed that many of the well-known algorithms of
discrete optimization are analogous to standard algorithms of the traditional
computational linear algebra.
|
math
|
2,773 |
Using noncommutative Groebner bases in solving partially prescribed matrix inverse completion problems
|
math.RA
|
We investigate the use of noncommutative Groebner bases in solving partially
prescribed matrix inverse completion problems. The types of problems considered
here are similar to those in [BLJW]. There the authors gave necessary and
sufficient conditions for the solution of a two by two block matrix completion
problem. Our approach is quite different from theirs and relies on symbolic
computer algebra.
Here we describe a general method by which all block matrix completion
problems of this type may be analyzed if sufficient computational power is
available. We also demonstrate our method with an analysis of all three by
three block matrix inverse completion problems with eleven blocks known and
seven unknown. We discover that the solutions to all such problems are of a
relatively simple form.
We then perform a more detailed analysis of a particular problem from the
31,824 three by three block matrix completion problems with eleven blocks known
and seven unknown. A solution to this problem of the form derived in [BLJW] is
presented.
Not only do we give a proof of our detailed result, but we describe the
strategy used in discovering our theorem and proof, since it is somewhat
unusual for these types of problems.
|
math
|
2,774 |
On the product by generators of characteristically nilpotent Lie S-algebras
|
math.RA
|
We introduce the product by generators of complex nilpotent Lie algebras,
which is a commutative product obtained from a central extension of the direct
sum of Lie algebras. We show that the product preserves also the characteristic
nilpotence provided that the multiplied algebras are $S$-algebras. In
particular, this shows the existence of nonsplit characteristically nilpotent
Lie algebras $\frak{h}$ such that the quotient $\frac{\dim \frak{h}-\dim
Z(\frak{h})}{\dim Z(\frak{h})} $ is as small as wanted.
|
math
|
2,775 |
An approach to Hopf algebras via Frobenius coordinates II
|
math.RA
|
We study a Hopf algebra $H$, which is finitely generated and projective over
a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions
in this setting, and provide a complete proof of Radford's formula for the
fourth power of the antipode, using Frobenius algebraic techniques. As further
applications, we extend Etingof and Gelaki's result that a separable and
coseparable Hopf algebra has antipode of order two, the result of Schneider
that Hopf subalgebras are twisted Frobenius extensions, and show that the
quantum double is always a Frobenius algebra.
|
math
|
2,776 |
Modulization and the enveloping ringoid
|
math.RA
|
Let A be an algebra in a variety V. We study the modulization of a pointed
A-overalgebra P, show that it is totally in any variety that P is totally in,
and apply this theory to the construction of the enveloping ringoid Z[A,V].
|
math
|
2,777 |
Are biseparable extensions Frobenius?
|
math.RA
|
In Secion~1 we describe what is known of the extent to which a separable
extension of unital associative rings is a Frobenius extension. A problem of
this kind is suggested by asking if three algebraic axioms for finite Jones
index subfactors are dependent. In Section~2 the problem in the title is
formulated in terms of separable bimodules.
In Section~3 we specialize the problem to ring extensions, noting that a
biseparable extension is a two-sided finitely generated projective, split,
separable extension. Some reductions of the problem are discussed and solutions
in special cases are provided. In Section~4 various examples are provided of
projective separable extensions that are neither finitely generated nor
Frobenius and which give obstructions to weakening the hypotheses of the
question in the title. We show in Section~5 that existing characterizations of
the separable extensions among the Frobenius extensions in are special cases of
a result for adjoint functors.
|
math
|
2,778 |
The Structure of the Inverse to the Sylvester Resultant Matrix
|
math.RA
|
Given polynomials a(z) of degree m and b(z) of degree n, we represent the
inverse to the Sylvester resultant matrix of a(z) and b(z), if this inverse
exists, as a canonical sum of m+n dyadic matrices each of which is a rational
function of zeros of a(z) and b(z). As a result, we obtain the polynomial
solutions X(z) of degree n-1 and Y(z) of degree m-1 to the equation
a(z)X(z)+b(z)Y(z)=c(z), where c(z) is a given polynomial of degree m+n-1, as
follows: X(z) is a Lagrange interpolation polynomial for the function c(z)/a(z)
over the set of zeros of b(z) and Y(z) is the one for the function c(z)/b(z)
over the set of zeros of a(z).
|
math
|
2,779 |
On an invariant related to a linear inequality
|
math.RA
|
Let A be an m-dimensional vector with positive real entries. Let A_{i,j} be
the vector obtained from A on deleting the entries A_i and A_j. We investigate
some invariant and near invariants related to the solutions E (m-2 dimensional
vectors with entries either +1 or -1) of the linear inequality |A_i-A_j| <
<E,A_{i,J}> < A_i+A_j, where <,> denotes the usual inner product. One of our
methods relates, by the use of Rademacher functions, integrals involving
trigonometric quantities to these quantities.
|
math
|
2,780 |
Generalized Projection Operators in Geometric Algebra
|
math.RA
|
Given an automorphism and an anti-automorphism of a semigroup of a Geometric
Algebra, then for each element of the semigroup a (generalized) projection
operator exists that is defined on the entire Geometric Algebra. A single
fundamental theorem holds for all (generalized) projection operators. This
theorem makes previous projection operator formulas equivalent to each other.
The class of generalized projection operators includes the familiar subspace
projection operation by implementing the automorphism `grade involution' and
the anti-automorphism `inverse' on the semigroup of invertible versors. This
class of projection operators is studied in some detail as the natural
generalization of the subspace projection operators. Other generalized
projection operators include projections onto any invertible element, or a
weighted projection onto any element. This last projection operator even
implies a possible projection operator for the zero element.
|
math
|
2,781 |
On deformations of the filiform Lie superalgebra $L_{n,m}$
|
math.RA
|
In this work, we recall that every filiform Lie superalgebra is a deformation
of the superalgebra $L_{n,m}$. We study the even cocycles which give this
nilpotent deformations. A family of independent bilinear maps will help us to
describe this cocycles. At the end an evaluation of the dimension of the space
$Z_0^2(L_{n,m},L_{n,m})$ is established.
|
math
|
2,782 |
Some Explicit Solutions of the Additive Deligne-Simpson Problem and Their Applications (Preprint)
|
math.RA
|
In this paper we construct three infinite series and two extra triples of
complex matrices B, C, and A=B+C of special spectral types associated to C.
Simpson's classification in his paper ``Products of Matrices'' and a
classification of multiple flag varieties with finitely many orbits of the
diagonal action of the general linear group by P. Magyar, J. Weyman, and A.
Zelevinsky. This enables us to construct Fuchsian systems of differential
equations which generalize the hypergeometric equation of Gauss-Riemann. In a
sense, they are the closest relatives of the famous equation, because their
triples of spectral flags have finitely many orbits of the diagonal action of
the general linear group in the space of solutions. We also construct a scalar
product such that A, B, and C are self-adjoint with respect to it. In the
context of Fuchsian systems, this scalar product becomes a monodromy invariant
complex symmetric bilinear form in the space of solutions.
When the eigenvalues of A, B, and C are real, the matrices and the scalar
product become real as well. We find inequalities on the eigenvalues of A, B,
and C which make the scalar product positive-definite.
As proved by A. Klyachko, the spectra of three hermitian (or real symmetric)
matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple
spectra. He also gave a recursive algorithm to generate inequalities describing
the cone. The inequalities we obtain describe non-recursively some faces of the
Klyachko cone.
|
math
|
2,783 |
Orthonormal Eigenbases over the Octonions
|
math.RA
|
We previously showed that the real eigenvalues of 3x3 octonionic Hermitian
matrices form two separate families, each containing 3 eigenvalues, and each
leading to an orthonormal decomposition of the identity matrix, which would
normally correspond to an orthonormal basis. We show here that it nevertheless
takes both families in order to decompose an arbitrary vector into components,
each of which is an eigenvector of the original matrix; each vector therefore
has 6 components, rather than 3.
|
math
|
2,784 |
Counting equivalence classes of irreducible representations
|
math.RA
|
Let $n$ be a positive integer, and let $R$ be a (possibly infinite
dimensional) finitely presented algebra over a computable field of
characteristic zero. We describe an algorithm for deciding (in principle)
whether $R$ has at most finitely many equivalence classes of $n$-dimensional
irreducible representations. When $R$ does have only finitely many such
equivalence classes, they can be effectively counted (assuming that $k[x]$
posesses a factoring algorithm).
|
math
|
2,785 |
Frobenius Functors of the second kind
|
math.RA
|
A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second
type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences
$\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide
examples of Frobenius pairs of the second kind. We study Frobenius pairs of the
second kind between categories of modules, comodules, and comodules over a
coring. We also show that a finitely generated projective Hopf algebra over a
commutative ring is always a Frobenius extension of the second kind, and prove
that the integral spaces of the Hopf algebra and its dual are isomorphic.
|
math
|
2,786 |
A note on the classification of naturally graded Lie algebras with linear characteristic sequence
|
math.RA
|
For sufficiently high dimensions, the naturally graded nonsplit nilpotent Lie
algebras with linear characteristic sequence are classified.
|
math
|
2,787 |
Simple roots of deformed preprojective algebras
|
math.RA
|
W. Crawley-Boevey has given a description of the set of dimension vectors of
simple representations of the deformed preprojective algebras. In this note we
give alternative descriptions. Note however that our descriptions depend on
irreducibility of the quotient varieties, so they do NOT give a shorter proof
of Crawley-Boevey's result.
|
math
|
2,788 |
Parametrization by polytopes of intersections of orbits by conjugation
|
math.RA
|
Let S be an nXn real symmetric matrix with spectral decomposition S=Q^T
Lambda Q, where Q is an orthogonal matrix and Lambda is diagonal with simple
spectrum {lambda_1,..., lambda_n}. Also let O_S e R_S be the orbits by
conjugation of S by, respectively, orthogonal matrices and upper triangular
matrices with positive diagonal. Denote by F_S the intersection O_S and R_S. We
show that the map F tha goes from the closure of F_S to R^n and takes S' =
(Q')^T Lambda Q' to diag(Q' Lambda (Q')^T) is a smooth bijection onto its range
P_S, the convex hull of some subset of the n! permuatations of (lambda_1, ...,
lambda_n). We also find necessary and sufficient conditions for P_S to have n!
vertices.
|
math
|
2,789 |
On gradings of matrix algebras and descent theory
|
math.RA
|
We classify gradings on matrix algebras by a finite abelian group. A grading
is called good if all elementary matrices are homogeneous. For cyclic groups,
all gradings on a matrix algebra over an algebraically closed field are good.
We can count the number of good gradings by a cyclic group. Using descent
theory, we classify non-good gradings on a matrix algebra that become good
after a base extension.
|
math
|
2,790 |
Simple completable contractions of nilpotent Lie algebras
|
math.RA
|
We study a certain class of non-maximal rank contractions of the nilpotent
Lie algebra $\frak{g}_{m}$ and show that these contractions are completable Lie
algebras. As a consequence a family of solvable complete Lie algebras of
non-maximal rank is given in arbitrary dimension.
|
math
|
2,791 |
Strongly graded hereditary orders
|
math.RA
|
Let R be a Dedekind domain with global quotient field K. The purpose of this
note is to provide a characterization of when a strongly graded R-order with
semiprime 1-component is hereditary. This generalizes earlier work by the first
author and G. Janusz (Trans. Amer. Math. Soc. 352 (2000), 3381-3410).
|
math
|
2,792 |
Applications of Perron-Frobenius Theory to Population Dynamics
|
math.RA
|
By the use of Perron-Frobenius theory, simple proofs are given of the
Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the
net reproductive rate occurring in matrix models of population dynamics. The
latter result is further refined with some additional nonnegative matrix
theory. When the fertility matrix is scaled by the net reproductive rate, the
growth rate of the model is 1. More generally, we show how to achieve a given
growth rate for the model by scaling the fertility matrix. Demographic
interpretations of the results are given.
|
math
|
2,793 |
Dualizing Complexes and Tilting Complexes over Simple Rings
|
math.RA
|
We prove that two-sided tilting complexes, and dualizing complexes, over
simple Goldie rings (with some technical conditions) are always shifts of
invertible bimodules. This allows us to describe the derived Picard groups of
such rings, and to deduce these are Gorenstein (and sometimes even
Auslander-Gorenstein Cohen-Macaulay) rings.
|
math
|
2,794 |
Automorphisms of tiled orders
|
math.RA
|
Let Lambda be a tiled R-order. We give a description of Aut_R(Lambda) as the
semidirect product of Inn(Lambda) and a certain subgroup of Aut(Q(Lambda)),
where Q(Lambda) is the link graph of Lambda. Additionally, we give criteria for
determining when an element of Aut(Q(Lambda)) belongs to this subgroup in terms
of the exponent matrix for Lambda.
|
math
|
2,795 |
Fields of definition for division algebras
|
math.RA
|
Let $A$ be a finite-dimensional division algebra containing a base field $k$
in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if
$A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that:
(1) In many cases $A$ can be defined over a rational extension of $k$. (2) If
$A$ has odd degree $n \ge 5$, then $A$ is defined over a field $F_0$ of
transcendence degree at most $(n-1)(n-2)/2$ over $k$. (3) If $A$ is a $Z/m
\times Z/2$-crossed product for some $m \ge 2$ (and in particular, if $A$ is
any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of
two symbol algebras. Consequently, $M_m(A)$ can be defined over a field $F_0$
of transcendence degree at most 4 over $k$. (4) If $A$ has degree 4 then the
trace form of $A$ can be defined over a field $F_0$ of transcendence degree at
most 4. (In (1), (3), and (4) we assume that the center of $A$ contains certain
roots of unity.)
|
math
|
2,796 |
$K_0$ of purely infinite simple regular rings
|
math.RA
|
We extend the notion of a purely infinite simple C*-algebra to the context of
unital rings, and we study its basic properties, specially those related to
K-Theory. For instance, if $R$ is a purely infinite simple ring, then
$K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated
projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by
adjoining a new zero element, and $K_1(R)$ is the abelianization of the group
of units of $R$. We develop techniques of construction, obtaining new examples
in this class in the case of von Neumann regular rings, and we compute the
Grothendieck groups of these examples. In particular, we prove that every
countable abelian group is isomorphic to $K_0$ of some purely infinite simple
regular ring. Finally, some known examples are analyzed within this framework.
|
math
|
2,797 |
Contractions and generalized Casimir invariants
|
math.RA
|
We prove that if $\frak{g}^{\prime}$ is a contraction of a Lie algebra
$\frak{g}$ then the number of functionally independent invariants of
$\frak{g}^{\prime}$ is at least that of $\frak{g}$. This allows to determine
explicitly the number of invariants of Lie algebras carrying a supplementary
structure, such as linear contact or linear forms whose differential is
symplectic.
|
math
|
2,798 |
Balanced d-lattices are complemented
|
math.RA
|
We show that all balanced d-lattices must be complemented, answering a
question of Chajda and Eigenthaler.
(A bounded lattice is balanced if any two congruences agree on their
1-classes iff they agree on their 0-classes.)
Our main tool is the characterization of d-lattices (a class of bounded
lattices including the bounded distributive lattices, originally defined by a
property of their compact congruences) as exactly those lattices in which all
maximal filters/ideals are prime.
|
math
|
2,799 |
Maschke functors, semisimple functors and separable functors of the second kind. Applications
|
math.RA
|
We introduce separable functors of the second kind (or $H$-separable
functors) and $H$-Maschke functors. $H$-separable functors are generalizations
of separable functors. Various necessary and sufficient conditions for a
functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir
elements (integrals, in the case of Hopf algebras), are given. An $H$-separable
functor is always $H$-Maschke, but the converse holds in particular situations.
A special role will be played by Frobenius functors and their relations to
$H$-separability. Our concepts are applied to modules, comodules, entwined
modules, quantum Yetter-Drinfeld modules, relative Hopf modules.
|
math
|
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