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2,600 |
The multigraded Nijenhuis-Richardson Algebra, its universal property and application
|
math.QA
|
We define two $(n+1)$ graded Lie brackets on spaces of multilinear mappings.
The first one is able to recognize $n$-graded associative algebras and their
modules and gives immediately the correct differential for Hochschild
cohomology. The second one recognizes $n$-graded Lie algebra structures and
their modules and gives rise to the notion of Chevalley cohomology.
|
math
|
2,601 |
Towards the Chern-Weil homomormism in non-commutative differential geometry
|
math.QA
|
In this short review article we sketch some developments which should
ultimately lead to the analogy of the Chern-Weil homomorphism for principal
bundles in the realm of non-commutative differential geometry. Principal
bundles there should have Hopf algebras as structure `cogroups'. Since the
usual machinery of Lie algebras, connection forms, etc\., just is not available
in this setting, we base our approach on the Fr\"olicher--Nijenhuis bracket.
|
math
|
2,602 |
A quantum-group-like structure on noncommutative 2-tori
|
math.QA
|
In this paper we show that in the case of noncommutative two-tori one gets in
a natural way simple structures which have analogous formal properties as Hopf
algebra structures but with a deformed multiplication on the tensor product.
|
math
|
2,603 |
Jaeger's Higman-Sims state model and the B_2 spider
|
math.QA
|
Jaeger [Geom. Dedicata 44 (1992), 23-52] discovered a remarkable checkerboard
state model based on the Higman-Sims graph that yields a value of the Kauffman
polynomial, which is a quantum invariant of links. We present a simple argument
that the state model has the desired properties using the combinatorial $B_2$
spider [Comm. Math. Phys. 180 (1996), 109-151].
|
math
|
2,604 |
Quantum and braided diffeomorphism groups
|
math.QA
|
We develop a general theory of `quantum' diffeomorphism groups based on the
universal comeasuring quantum group $M(A)$ associated to an algebra $A$ and its
various quotients. Explicit formulae are introduced for this construction, as
well as dual quasitriangular and braided R-matrix versions. Among the examples,
we construct the $q$-diffeomorphisms of the quantum plane $yx=qxy$, and recover
the quantum matrices $M_q(2)$ as those respecting its braided group addition
law.
|
math
|
2,605 |
Hamiltonian Reduction and the Construction of q-Deformed Extensions of the Virasoro Algebra
|
math.QA
|
In this paper we employ the construction of Dirac bracket for the remaining
current of $sl(2)_q$ deformed Kac-Moody algebra when constraints similar to
those connecting the $sl(2)$-WZW model and the Liouville theory are imposed and
show that it satisfy the q-Virasoro algebra proposed by Frenkel and
Reshetikhin. The crucial assumption considered in our calculation is the
existence of a classical Poisson bracket algebra induced, in a consistent
manner by the correspondence principle, mapping the quantum generators into
commuting objects of classical nature preserving their algebra.
|
math
|
2,606 |
On Gauss decomposition of quantum groups and Jimbo homomorphism
|
math.QA
|
It is shown that the properties of the Gauss decomposition of quantum groups
and the known Jimbo homomorphism permit us to realize these groups as
subalgebras of well defined algebras constructed from generators of the
corresponding undeformed Lie algebras.
|
math
|
2,607 |
"Wick Rotations": The Noncommutative Hyperboloids, and other surfaces of rotations
|
math.QA
|
A ``Wick rotation'' is applied to the noncommutative sphere to produce a
noncommutative version of the hyperboloids. A harmonic basis of the associated
algebra is given. It is noted that, for the one sheeted hyperboloid, the vector
space for the noncommutative algebra can be completed to a Hilbert space, where
multiplication is not continuous. A method of constructing noncommutative
analogues of surfaces of rotation, examples of which include the paraboloid and
the $q$-deformed sphere, is given. Also given are mappings between
noncommutative surfaces, stereographic projections to the complex plane and
unitary representations. A relationship with one dimensional crystals is
highlighted.
|
math
|
2,608 |
Some examples of quantum groups in higher genus
|
math.QA
|
This is a survey of our construction of current algebras, associated with
complex curves and rational differentials. We also study in detail two classes
of examples. The first is the case of a rational curve with differentials $z^n
dz$; these algebras are ``building blocks'' for the quantum current algebras
introduced in our earlier work. The second is the case of a genus $>1$ curve
$X$, endowed with a regular differential having only double zeroes.
|
math
|
2,609 |
Quantization of Lie bialgebras, IV
|
math.QA
|
This paper is a continuation of "Quantization of Lie bialgebras, III"
(q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra
F(R)_\z associated to a quantum R-matrix R(z) with a spectral parameter, and a
set of points \z=(z_1,...,z_n). This algebra is generated by entries of a
matrix power series T_i(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan
type commutation relations, and is a quantization of the group GL_N[[t]].
In this paper we consider the quotient F_0(R)_\z of F(R)_\z by the relations
\qdet_R(T_i)=1, where \qdet_R is the quantum determinant associated to R (for
rational, trigonometric, or elliptic R-matrices). This is also a Hopf algebra,
which is a quantization of the group SL_N[[t]].
This paper was inspired by the pioneering paper of I.Frenkel and Reshetikhin.
The main goal of this paper is to study the representation theory of the
algebra F_0(R)_\z and of its quantum double, and show how the consideration of
coinvariants of this double (quantum conformal blocks) naturally leads to the
quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin.
Our construction for the rational R-matrix is a quantum analogue of the
standard derivation of the Knizhnik-Zamolodchikov equations in the
Wess-Zumino-Witten model of conformal field theory, and for the elliptic
R-matrix is a quantum analogue of the construction of Kuroki and Takebe.
Our result is a generalization of the construction of Enriques and Felder,
which appeared while this paper was in preparation. Enriques and Felder gave a
derivation of the quantum KZ equations from coinvariants in the case of the
rational R-matrix and N=2.
|
math
|
2,610 |
Set-theoretical solutions to the quantum Yang-Baxter equation
|
math.QA
|
In 1992 V.Drinfeld formulated a number of problems in quantum group theory.
In particular, he suggested to consider ``set-theoretical'' solutions to the
quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the
set $X\times X$, where X is a fixed set. In this paper we study such solutions,
which in addition satisfy the unitarity and nondegeneracy conditions. We
discuss the geometric and algebraic interpretations of such solutions,
introduce several constructions of them, and make first steps towards their
classification.
|
math
|
2,611 |
The Aarhus integral of rational homology 3-spheres II: Invariance and universality
|
math.QA
|
We continue the work started in part I (q-alg/9706004) and prove the
invariance and universality in the class of finite type invariants of the
object defined and motivated there, namely the Aarhus integral of rational
homology 3-spheres. Our main tool in proving invariance is a translation scheme
that translates statements in multi-variable calculus (Gaussian integration,
integration by parts, etc.) to statements about diagrams. Using this scheme the
straight-forward "philosophical" calculus-level proofs of part I become
straight-forward honest diagram-level proofs here. The universality proof is
standard and utilizes a simple "locality" property of the Kontsevich integral.
|
math
|
2,612 |
Infinite Hopf family of elliptic algebras and bosonization
|
math.QA
|
Elliptic current algebras E_{q,p}(\hat{g}) for arbitrary simply laced finite
dimensional Lie algebra g are defined and their co-algebraic structures are
studied. It is shown that under the Drinfeld like comultiplications, the
algebra E_{q,p}(\hat{g}) is not co-closed for any g. However putting the
algebras E_{q,p}(\hat{g}) with different deformation parameters together, we
can establish a structure of infinite Hopf family of algebras. The level 1
bosonic realization for the algebra E_{q,p}(\hat{g}) is also established.
|
math
|
2,613 |
Annihilating ideals and tilting functors
|
math.QA
|
We use Kazhdan-Lusztig tensoring to, first, describe annihilating ideals of
highest weight modules over an affine Lie algebra in terms of the corresponding
VOA and, second, to classify tilting functors, an affine analogue of projective
functors known in the case of a simple Lie algebra. For the sake of
completeness, the classification of annihilating ideals is borrowed from our
previous work, q-alg/9711011; the part on tilting functors is new.
|
math
|
2,614 |
A General $q$-Oscillator Algebra
|
math.QA
|
It is well-known that the Macfarlane-Biedenharn $q$-oscillator and its
generalization has no Hopf structure, whereas the Hong Yan $q$-oscillator can
be endowed with a Hopf structure. In this letter, we demonstrate that it is
possible to construct a general $q$-oscillator algebra which includes the
Macfarlane-Biedenharn oscillator algebra and the Hong Yan oscillator algebra as
special cases.
|
math
|
2,615 |
Quantized W-algebra of ${\frak sl}(2,1)$ : a construction from the quantization of screening operators
|
math.QA
|
Starting from bosonization, we study the operator that commute or commute
up-to a total difference with of any quantized screen operator of a free field.
We show that if there exists a operator in the form of a sum of two vertex
operators which has the simplest correlation functions with the quantized
screen operator, namely a function with one pole and one zero, then, the screen
operator and this operator are uniquely determined, and this operator is the
quantized virasoro algebra. For the case when the screen is a fermion, there
are a family of this kind of operator, which give new algebraic structures.
Similarly we study the case of two quantized screen operator, which uniquely
gives us the quantized W-algebra corresponding to $sl(3)$ for the generic case,
and a new algebra, which is a quantized W-algebra corresponding to ${\frak
sl}(2,1)$, for the case that one of the two screening operators is or both are
fermions.
|
math
|
2,616 |
Partial Gauss decomposition, \bf $U_q(\widehat{\frak{gl}(n-1)})\in U_q(\widehat{\frak{gl}(n)}) $ and Zamolodchikov algebra
|
math.QA
|
We use the idea of partial Gauss decomposition to study structures related to
$U_q(\widehat{{{\frak{gl}}}(n-1)})$ inside $U_q(\widehat{{{\frak{gl}}}(n)}) $.
This gives a description of $U_q(\widehat{{{\frak{gl}}}(n)})$ as an extension
of $U_q(\widehat{{{\frak{gl}}}(n-1)})$ with Zamolodchikov algebras, We explain
the connection of this new realization with form factors.
|
math
|
2,617 |
n-ary Lie and Associative Algebras
|
math.QA
|
With the help of the multigraded Nijenhuis-- Richardson bracket and the
multigraded Gerstenhaber bracket from [7] for every $n\ge 2$ we define $n$-ary
associative algebras and their modules and also $n$-ary Lie algebras and their
modules, and we give the relevant formulas for Hochschild and Chevalley
cohomogy.
|
math
|
2,618 |
Cyclic operads and homology of graph complexes
|
math.QA
|
We will consider P-graph complexes, where P is a cyclic operad. P-graph
complexes are natural generalizations of Kontsevich's graph complexes -- for P
= the operad for associative algebras it is the complex of ribbon graphs, for P
= the operad for commutative associative algebras, the complex of all graphs.
We construct a `universal class' in the cohomology of the graph complex with
coefficients in a theory. The Kontsevich-type invariant is then an evaluation,
on a concrete cyclic algebra, of this class. We also explain some results of M.
Penkava and A. Schwarz on the construction of an invariant from a cyclic
deformation of a cyclic algebra. Our constructions are illustrated by a `toy
model' of tree complexes.
|
math
|
2,619 |
A link between two elliptic quantum groups
|
math.QA
|
We construct a fully faithful functor from the category C_F of
finite-dimensional representations of Felder's (dynamical) elliptic quantum
group E_{tau,gamma}(gl(n)) to a cretain category D_B of (infinite-dimensional)
representations of Belavin's quantum elliptic algebra B by difference
operators, and a fully faithful functor from the category C_B of
finite-dimensional representations of B to D_B. As a corollary, we show that
the abelian subcategories of C_B and C_F generated by tensor products of vector
representations are equivalent.
|
math
|
2,620 |
Lifting formulas II
|
math.QA
|
In the present paper we generalize the lifting formulas from [Sh] and obtain
the formula for the (2n+2l-1)-cocycle on the LIe algebra of differential
operators on a n-dimensional space for arbitrary n and l.
|
math
|
2,621 |
On Semisimple Hopf algebras of dimension pq
|
math.QA
|
In this paper we consider some properties of semisimple Hopf algebras of
dimension pq where p and q are distinct primes. These properties are useful for
classification of such Hopf algebras. In particular, we show that for such a
Hopf algebra H, if H and H^* are of Frobenius type then H or H^* is a group
algebra.
|
math
|
2,622 |
Semisimple Hopf algebras of dimension pq are trivial
|
math.QA
|
Masuoka proved that for a prime p, semisimple Hopf algebras of dimension 2p
over an algebraically closed field k of characteristic 0, are trivial (i.e. are
either group algebras or the dual of group algebras). Westreich and the second
author obtained the same result for dimension 3p, and then pushed the analysis
further and among the rest obtained the same result for semisimple Hopf
algebras H of dimension pq so that H and H^* are of Frobenius type (i.e. the
dimensions of their irreducible representations divide the dimension of H).
They concluded with the conjecture that any semisimple Hopf algebra H of
dimension pq over k, is trivial. In this paper we use Theorem 1.4 in our
previous paper q-alg/9712033 to prove that both H and H^* are of Frobenius
type, and hence prove this conjecture.
|
math
|
2,623 |
On pointed Hopf algebras and Kaplansky's 10th conjecture
|
math.QA
|
In this paper we construct and study two new families of finite dimensional
pointed Hopf algebras which generalize Radford's families. We show that over
any infinite field which contains a primitive nth root of unity, one of the
families contains infinitely many non-isomorphic Hopf algebras of any dimension
of the form Nn^2, where 2<n<N are integers so that n divides N. We thus answer
in the negative Kaplansky's 10th conjecture from 1975 on the finite number of
types of Hopf algebras of a given dimension.
|
math
|
2,624 |
Exchange dynamical quantum groups
|
math.QA
|
For any simple Lie algebra g and any complex number q which is not zero or a
nontrivial root of unity, we construct a dynamical quantum group (Hopf
algebroid), whose representation theory is essentially the same as the
representation theory of the quantum group U_q(g). This dynamical quantum group
is obtained from the fusion and exchange relations between intertwining
operators in representation theory of U_q(g), and is an algebraic structure
standing behind these relations.
|
math
|
2,625 |
A quantum octonion algebra
|
math.QA
|
Using the natural irreducible 8-dimensional representation and the two spin
representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a
quantum analogue of the split octonions and study its properties. We prove that
the quantum octonion algebra satisfies the q-Principle of Local Triality and
has a nondegenerate bilinear form which satisfies a q-version of the
composition property. By its construction, the quantum octonion algebra is a
nonassociative algebra with a Yang-Baxter operator action coming from the
R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a
$U_q$(D$_4$)-module homomorphism. Using that, we prove identities for the
quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation
theory'' proofs for very well-known identities satisfied by the octonions. In
the process of constructing the quantum octonions we introduce an algebra which
is a q-analogue of the 8-dimensional para-Hurwitz algebra.
|
math
|
2,626 |
t-deformation of quantum Schubert polynomials
|
math.QA
|
We construct a certain solution to the Witten--Dijkgraf--Verlinde--Verlinde
equation related to the small quantum cohomology ring of flag variety, and
study the t-deformation of quantum Schubert polynomials corresponding to this
solution.
|
math
|
2,627 |
A Mayer-Vietoris Theorem for the Kauffman Bracket Skein Module
|
math.QA
|
The nth relative Kauffman bracket skein modules are defined and two theorems
are given relating them to the Kauffman bracket skein module of a 3-manifold.
The first theorem covers the case when the 3-manifold is split along a
separating closed orientable surface and the second theorem addresses the case
when the surface is nonseparating.
|
math
|
2,628 |
Quantum Kac-Moody Algebras and Vertex Representations
|
math.QA
|
We introduce an affinization of the quantum Kac-Moody algebra associated to a
symmetric generalized Cartan matrix. Based on the affinization, we construct a
representation of the quantum Kac-Moody algebra by vertex operators from
bosonic fields. We also obtain a combinatorial indentity about Hall-Littlewood
polynomials.
|
math
|
2,629 |
Yangian actions on higher level irreducible integrable modules of affine gl(N)
|
math.QA
|
An action of the Yangian of the general Lie algebra gl(N) is defined on every
irreducible integrable highest weight module of affine gl(N) with level greater
than 1. This action is derived, by means of the Drinfeld duality and a
subsequent semi-infinite limit, from a certain induced representation of the
degenerate double affine Hecke algebra H. Each vacuum module of affine gl(N) is
decomposed into irreducible Yangian subrepresentations by means of the
intertwiners of H. Components of this decomposition are parameterized by
semi-infinite skew Young diagrams.
|
math
|
2,630 |
Braided Hopf algebras over non abelian finite groups
|
math.QA
|
This is a survey of general aspects of the theory of braided Hopf algebras
with emphasis on a special class of braided graded Hopf algebras named tobas.
The interest on tobas arises from problems of classification of pointed Hopf
algebras. We discuss tobas from different points of view following ideas of
Lusztig, Nichols and Schauenburg. We then concentrate on braided Hopf algebras
in the Yetter-Drinfeld category over H, where H is the group algebra of a non
abelian finite group. We present some finite dimensional examples arising in an
unpublished work by Milinski and Schneider.
|
math
|
2,631 |
Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras
|
math.QA
|
We classify all 4-dimensional first order bicovariant calculi on the
Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order
bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we
assume that the bicovariant bimodules are generated as left modules by the
differentials of the quantum group generators. It is found that there are 3
1-parameter families of 4-dimensional bicovariant first order calculi on
GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant
calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a
classical-like reduction from any one of the three families of 4-dimensional
calculi on GL_{h,g}(2). Details of the higher order calculi and also the
quantum Lie algebras are presented for all calculi. The quantum Lie algebra
obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic
to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian
universal enveloping algebra U_{h}(sl(2)) and also through a consideration of
the decomposition of the tensor product of two copies of the deformed adjoint
module. We also obtain the quantum Killing form for this quantum Lie algebra.
|
math
|
2,632 |
Quantized flag manifolds and irreducible *-representations
|
math.QA
|
We study irreducible *-representations of a certain quantization of the
algebra of polynomial functions on a generalized flag manifold regarded as a
real manifold. All irreducible *-representations are classified for a subclass
of flag manifolds containing in particular the irreducible compact Hermitian
symmetric spaces. For this subclass it is shown that the irreducible
*-representations are parametrized by the symplectic leaves of the underlying
Poisson bracket. We also discuss the relation between the quantized flag
manifolds studied in this paper and the quantum flag manifolds studied by
Soibelman, Lakshimibai and Reshetikhin, Jurco and Stovicek, and Korogodsky.
|
math
|
2,633 |
Differential Calculus on Quantum Spheres
|
math.QA
|
We study covariant differential calculus on the quantum spheres S_q^2N-1.
Two classification results for covariant first order differential calculi are
proved. As an important step towards a description of the noncommutative
geometry of the quantum spheres, a framework of covariant differential calculus
is established, including a particular first order calculus obtained by
factorization, higher order calculi and a symmetry concept.
|
math
|
2,634 |
Quasialgebra structure of the octonions
|
math.QA
|
We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x
Z_2 in the quasitensor category of representations of a quasi-Hopf algebra
associated to a group 3-cocycle. We consider general quasi-associative algebras
of this type and some general constructions for them, including quasi-linear
algebra and representation theory, and an automorphism quasi-Hopf algebra.
Other examples include the higher 2^n-onion Cayley algebras and examples
associated to Hadamard matrices.
|
math
|
2,635 |
Level One Representations of Quantum Affine Algebras $U_q(C^{(1)}_n)$
|
math.QA
|
We give explicit constructions of quantum symplectic affine algebras at level
1 using vertex operators.
|
math
|
2,636 |
New combinatorial formula for modified Hall-Littlewood polynomials
|
math.QA
|
We obtain new combinatorial formulae for modified Hall--Littlewood
polynomials, for matrix elements of the transition matrix between the
elementary symmetric functions and Hall-Littlewood's ones, and for the number
of rational points over the finite field of unipotent partial flag variety. The
definitions and examples of generalized mahonian statistic on the set of
transport matrices and dual mahonian statistic on the set of transport
(0,1)-matrices are given. We also review known q-analogues of
Littlewood-Richardson numbers and consider their possible generalizations.
Several conjectures about multinomial fermionic formulae for homogeneous
unrestricted one dimensional sums and generalized Kostka-Foulkes polynomials
are formulated. Finally we suggest two parameters deformations of polynomials
$P_{\lambda\mu}(t)$ and one dimensional sums.
|
math
|
2,637 |
Drinfeldians
|
math.QA
|
We construct two-parameter deformation of an universal enveloping algebra
$U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a
finite-dimensional complex simple Lie algebra (or superalgebra). This new
quantum Hopf algebra called the Drinfeldian $D_{q\eta}(g)$ can be considered as
a quantization of $U(g[u])$ in the direction of a classical r-matrix which is a
sum of the simple rational and trigonometric r-matrices. The Drinfeldian
$D_{q\eta}(g)$ contains $U_{q}(g)$ as a Hopf subalgebra, moreover $U_{q}(g[u])$
and $Y_{\eta}(g)$ are its limit quantum algebras when the $D_{q\eta}(g)$
deformation parameters $\eta$ goes to 0 and $q$ goes to 1, respectively. These
results are easy generalized to a supercase, i.e. when $g$ is a
finite-dimensional contragredient simple superalgebra.
|
math
|
2,638 |
Cohomology of Conformal Algebras
|
math.QA
|
Conformal algebra is an axiomatic description of the operator product
expansion of chiral fields in conformal field theory. On the other hand, it is
an adequate tool for the study of infinite-dimensional Lie algebras satisfying
the locality property. The main examples of such Lie algebras are those
``based'' on the punctured complex plane, like the Virasoro algebra and loop
algebras. In the present paper we develop a cohomology theory of conformal
algebras with coefficients in an arbitrary module. It possesses standard
properties of cohomology theories; for example, it describes extensions and
deformations. We offer explicit computations for most of the important
examples.
|
math
|
2,639 |
On the decomposition matrices of the quantized Schur algebra
|
math.QA
|
We prove the decomposition conjecture of Leclerc and Thibon for the Schur
algebra. We also give a new approach to the Lusztig conjecture for the
dimension of the simple U(sl_k)-modules at roots of unity via canonical bases
of the Hall algebra.
|
math
|
2,640 |
Another proof of M. Kontsevich formality theorem
|
math.QA
|
The paper contains an alternative proof of M. Kontsevich Formality Theorem.
|
math
|
2,641 |
On the signature of certain intersection forms
|
math.QA
|
We prove a conjecture of Zuber on the signature of intersection froms
associated with affine algebras of type A.
|
math
|
2,642 |
q-deformed Hermite Polynomials in q-Quantum Mechanics
|
math.QA
|
The q-special functions appear naturally in q-deformed quantum mechanics and
both sides profit from this fact. Here we study the relation between the
q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss:
recursion formula, generating function, Christoffel-Darboux identity,
orthogonality relations and the moment functional.
|
math
|
2,643 |
Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^3
|
math.QA
|
We propose the following principle to study pointed Hopf algebras, or more
generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a
Hopf algebra A, consider its coradical filtration and the associated graded
coalgebra grad(A). Then grad(A) is a graded Hopf algebra, since the coradical
A_0 of A is a Hopf subalgebra. In addition, there is a projection \pi: grad(A)
\to A_0; let R be the algebra of coinvariants of \pi. Then, by a result of
Radford and Majid, R is a braided Hopf algebra and grad(A) is the bosonization
(or biproduct) of R and A_0: grad(A) is isomorphic to (R # A_0). The principle
we propose to study A is first to study R, then to transfer the information to
grad(A) via bosonization, and finally to lift to A. In this article, we apply
this principle to the situation when R is the simplest braided Hopf algebra: a
quantum linear space. As consequences of our technique, we obtain the
classification of pointed Hopf algebras of order p^3 (p an odd prime) over an
algebraically closed field of characteristic zero; with the same hypothesis,
the characterization of the pointed Hopf algebras whose coradical is abelian
and has index p or p^2; and an infinite family of pointed, non-isomorphic, Hopf
algebras of the same dimension. This last result gives a negative answer to a
conjecture of I. Kaplansky.
|
math
|
2,644 |
On generalized Abelian deformations
|
math.QA
|
We study sun-products on $\R^n$, i.e. generalized Abelian deformations
associated with star-products for general Poisson structures on $\R^n$. We show
that their cochains are given by differential operators. As a consequence, the
weak triviality of sun-products is established and we show that strong
equivalence classes are quite small. When the Poisson structure is linear
(i.e., on the dual of a Lie algebra), we show that the differentiability of
sun-products implies that covariant star-products on the dual of any Lie
algebra are equivalent each other.
|
math
|
2,645 |
A generalization of the Kostka-Foulkes polynomials
|
math.QA
|
Combinatorial objects called rigged configurations give rise to q-analogues
of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials
and two-column Macdonald-Kostka polynomials occur as special cases.
Conjecturally these polynomials coincide with the Poincare polynomials of
isotypic components of certain graded GL(n)-modules supported in a nilpotent
conjugacy class closure in gl(n).
|
math
|
2,646 |
The positive part of the quantized universal enveloping algebra of type A_n as a braided quantum group
|
math.QA
|
A generalized Hopf algebra structure for the positive (negative) part of the
Drinfeld-Jimbo quantum group of type A_n is established without make any use of
the usual deformation of the abelian part of sl_{n+1}.
|
math
|
2,647 |
Representations of the Generalized Lie Algebra sl(2)_q
|
math.QA
|
We construct finite-dimensional irreducible representations of two quantum
algebras related to the generalized Lie algebra $\ssll (2)_q$ introduced by
Lyubashenko and the second named author. We consider separately the cases of
$q$ generic and $q$ at roots of unity. Some of the representations have no
classical analog even for generic $q$. Some of the representations have no
analog to the finite-dimensional representations of the quantised enveloping
algebra $U_q(sl(2))$, while in those that do there are different matrix
elements.
|
math
|
2,648 |
On position operator spectral measure for deformed oscillator in the case of indetermine Hamburger moment problem
|
math.QA
|
The spectral measure of the position (momentum) operator $X$ for $q$-deformed
oscillator is calculated in the case of the indetermine Hamburger moment
problem. The exposition is given for concrete choice of generators for
$q$-oscillator algebra, although developed technique apply for every other
cases with indetermine moment problem. The Stieltjes transformation $m(z)$ of
spectral measure is expressed in terms of the entries of Jacobi matrix $X$
only. The direct connection between values of parameters labeling the spectral
measures and related selfadjoint extensions of $X$ is established.
|
math
|
2,649 |
Tressages des groupe de Poisson formels à dual quasitriangulaire
|
math.QA
|
Let $ \mathfrak{g} $ be a quasitriangular Lie bialgebra over a field $ K $ of
characteristic zero, and let $ \mathfrak{g}^* $ be its dual Lie bialgebra. We
prove that the formal Poisson group $ K\big[\big[\mathfrak{g}^*\big]\big] $ is
a braided Hopf algebra, thus generalizing a result due to Reshetikhin (in the
case $ \, \mathfrak{g} = \mathfrak{sl}(2,K) \, $). The proof is via quantum
groups, using the existence of a quasitriangular quantization of $
\mathfrak{g}^* $, as well as the fact that this one provides also a
quantization of $ K\big[\big[\mathfrak{g}^*\big]\big] \, $.
|
math
|
2,650 |
Monstrous Moonshine of higher weight
|
math.QA
|
We determine the space of 1-point correlation functions associated with the
Moonshine module: they are precisely those modular forms of non-negative
integral weight which are holomorphic in the upper half plane, have a pole of
order at most 1 at infinity, and whose Fourier expansion has constant 0. There
are Monster-equivariant analogues in which one naturally associates to each
element in the Monster a modular form of fixed weight k, the case k=0
corresponding to the original ``Moonshine'' of Conway and Norton.
|
math
|
2,651 |
On the Cohomology Ring of an Algebra
|
math.QA
|
We define several versions of the cohomology ring of an associative algebra.
These ring structures unify some well known operations from homological algebra
and differential geometry. They have some formal resemblance with the quantum
multiplication on Floer cohomology of free loop spaces. We discuss some
examples, as well as applications to index theorems, characteristic classes and
deformations.
|
math
|
2,652 |
Fukaya Type Categories for Associative Algebras
|
math.QA
|
We define for an associative algebra an $A_{\infty}$ category whose objects
are automorphisms of this algebra. This construction has some resemblance with
Fukaya'a categories related to Floer cohomology.
|
math
|
2,653 |
Tensor Operators for Uh(sl(2))
|
math.QA
|
Tensor operators for the Jordanian quantum algebra Uh(sl(2)) are considered.
Some explicit examples of them, which are obtained in the boson or fermion
realization, are given and their properties are studied. It is also shown that
the Wigner-Eckart's theorem can be extended to Uh(sl(2)).
|
math
|
2,654 |
Double quantization of $\cp$ type orbits by generalized Verma modules
|
math.QA
|
It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra
${\bf g}$ are equiped with a Poisson pencil generated by the
Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to
the "canonical" R-matrix. We realize quantization of this Poisson pencil on
$\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $
CP^n$) by means of q-deformed Verma modules.
|
math
|
2,655 |
Universal R-matrix for esoteric quantum group
|
math.QA
|
The universal $R$-matrix for a class of esoteric (non-standard) quantum
groups ${\cal U}_q(gl(2N+1))$ is constructed as a twisting of the universal
$R$-matrix ${\cal R}_S$ of the Drinfeld-Jimbo quantum algebras. The main part
of the twisting element ${\cal F}$ is chosen to be the canonical element of
appropriate pair of separated Hopf subalgebras (quantized Borel's ${\cal B}(N)
\subset {\cal U}_q(gl(2N+1))$), providing the factorization property of ${\cal
F}$. As a result, the esoteric quantum group generators can be expressed in
terms of the Drinfeld-Jimbo ones.
|
math
|
2,656 |
Twisting cocycles in fundamental representation and triangular bicrossproduct Hopf algebras
|
math.QA
|
We find the general solution to the twisting equation in the tensor bialgebra
$T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of
fundamental representation for a universal enveloping Lie algebra and its
quantum deformations. We suggest a procedure of constructing twisting cocycles
belonging to a given quasitriangular subbialgebra ${\cal H}\subset T({\bf R})$.
This algorithm generalizes Reshetikhin's approach, which involves cocycles
fulfilling the Yang-Baxter equation. Within this framework we study a class of
quantized inhomogeneous Lie algebras related to associative rings in a certain
way, for which we build twisting cocycles and universal $R$-matrices. Our
approach is a generalization of the methods developed for the case of
commutative rings in our recent work including such well-known examples as
Jordanian quantization of the Borel subalgebra of $sl(2)$ and the null-plane
quantized Poincar\'e algebra by Ballesteros at al. We reveal the role of
special group cohomologies in this process and establish the bicrossproduct
structure of the examples studied.
|
math
|
2,657 |
A cyclage poset structure for Littlewood-Richardson tableaux
|
math.QA
|
A graded poset structure is defined for the sets of Littlewood-Richardson
(LR) tableaux that count the multiplicity of an irreducible GL(n)-module in the
tensor product of irreducibles indexed by a sequence of rectangular partitions.
This poset generalizes the cyclage poset on column-strict tableaux defined by
Lascoux and Schutzenberger, and its grading function generalizes the charge
statistic. It is shown that the polynomials obtained by enumerating LR tableaux
by shape and the generalized charge, are the Poincare polynomials of isotypic
components of the certain modules supported in the closure of a nilpotent
conjugacy class.
|
math
|
2,658 |
The Two-Dimensional Quantum Galilei Groups
|
math.QA
|
The Poisson structures on two-dimensional Galilei group, classified in the
author previous paper are quantized. The dual quantum Galilei Lie algebras are
found.
|
math
|
2,659 |
Generalized quantum current algebras
|
math.QA
|
Two general families of new quantum deformed current algebras are proposed
and identified both as infinite Hopf family of algebras, a structure which
enable one to define ``tensor products'' of these algebras. The standard
quantum affine algebras turn out to be a very special case of both algebra
families, in which case the infinite Hopf family structure degenerates into
standard Hopf algebras. The relationship between the two algebra families as
well as their various special examples are discussed, and the free boson
representation is also considered.
|
math
|
2,660 |
On q-analogues of Riemann's zeta
|
math.QA
|
In the paper, we introduce $q$-deformations of the Riemann zeta function,
extend them to the whole complex plane, and establish certain estimates of the
number of roots. The construction is based on the recent difference
generalization of the Harish-Chandra theory of zonal spherical functions. We
also discuss numerical results, which indicate that the location of the zeros
of the $q$-zeta functions is far from random.
|
math
|
2,661 |
Calculating zeros of a q-zeta function numerically
|
math.QA
|
The note is a continuation of the previous paper ``On q-analogues of
Riemann's zeta'' (math.QA/980499). It contains an output of the computer
program calculating the zeros of the ``sharp'' q-zeta function.
|
math
|
2,662 |
Quantum Galois theory for compact Lie groups
|
math.QA
|
We establish a quantum Galois correspondence for compact Lie groups of
automorphisms acting on a simple vertex operator algebra.
|
math
|
2,663 |
A note on the generalised Lie algebra sl(2)q
|
math.QA
|
In a recent paper, V. Dobrev and A. Sudbery classified the highest-weight and
lowest-weight finite dimensional irreducible representations of the quantum Lie
algebra sl(2)_q introduced by V. Lyubashenko and A. Sudbery. The aim of this
note is to add to this classification all the finite dimensional irreducible
representations which have no highest weight and/or no lowest weight, in the
case when q is a root of unity. For this purpose, we give a description of the
enlarged centre.
|
math
|
2,664 |
Affine Weyl groups, discrete dynamical systems and Painleve equations
|
math.QA
|
A new class of representations of affine Weyl groups on rational functions
are constructed, in order to formulate discrete dynamical systems associated
with affine root systems. As an application, some examples of difference and
differential systems of Painleve type are discussed.
|
math
|
2,665 |
Induction of quantum group representations
|
math.QA
|
Induced representations for quantum groups are defined starting from
coisotropic quantum subgroups and their main properties are proved. When the
coisotropic quantum subgroup has a suitably defined section such
representations can be realized on associated quantum bundles on general
embeddable quantum homogeneous spaces.
|
math
|
2,666 |
Weyl group extension of quantized current algebras
|
math.QA
|
In this paper, we extend the Drinfeld current realization of quantum affine
algebras $U_q(\hat {\gg})$ and of the Yangians in several directions: we
construct current operators for non-simple roots of ${\gg}$, define a new braid
group action in terms of the current operators and describe the universal
R-matrix for the corresponding ``Drinfeld'' comultiplication in the form of
infinite product and in the form of certain integrals over current operators.
|
math
|
2,667 |
On the FRTS approach to quantized current algebras
|
math.QA
|
We study the possibility to establish $L$-operator's formalism by
Faddeev-Reshetikhin-Takhtajan-Semenov-Tian-Shansky (FRST) for quantized current
algebras, that is, for quantum affine algebras in the ''new realization '' by
V. Drinfeld with the corresponding Hopf algebra structure and for their Yangian
counterpart. We establish this formalism using the twisting procedure by
Tolstoy and the second author and explain the problems which FRST approach
encounter for quantized current algebras. We show also that, for the case of
$U_q(\hat {\frak sl}_n)$, entries of the L-operators of FRTS type give the
Drinfeld current operators for the non-simple roots, which we discovered
recently. As an application we deduce the commutation relations between these
current operators for $U_q(\hat {\frak sl}_3)$.
|
math
|
2,668 |
On the Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces
|
math.QA
|
Let A be a coquasitriangular Hopf algebra and X the subalgebra of A generated
by a row of a matrix corepresentation u or by a row of u and a row of the
contragredient representation u^c. In the paper left-covariant first order
differential calculi on the quantum group A are constructed and the
corresponding induced calculi on the left quantum space X are described. The
main tool for these constructions are the L-functionals associated with u. The
results are applied to the quantum homogeneous space GL_q(N)/GL_q(N-1).
|
math
|
2,669 |
The $ R $--matrix action of untwisted affine quantum groups at roots of 1
|
math.QA
|
Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra. The quantum
group $U_h(\hat{\frak g})$ (over $\mathbb{C}[[h]]$) is known to be a
quasitriangular Hopf algebra: in particular, it has a universal $ R $--matrix,
which yields an $ R $--matrix for each pair of representations of
$U_h(\hat{\frak g})$. On the other hand, the quantum group $U_q(\hat{\frak g})$
(over $\mathbb{C}(q) $) also has an $ R $--matrix for each pair of
representations, but it has not a universal $ R $--matrix so that one cannot
say that it is quasitriangular. Following Reshetikin, one introduces the
(weaker) notion of braided Hopf algebra: then $ U_q(\hat{\frak g})$ is a
braided Hopf algebra.
In this work we prove that also the unrestricted specializations of
$U_q(\hat{\frak g})$ at roots of 1 are braided: in particular, specializing $q$
at 1 we have that the function algebra $F \big[ \hat{H} \big]$ of the Poisson
proalgebraic group $\hat{H}$ dual of $\hat{G}$ (a Kac-Moody group with Lie
algebra $\hat{\frak g} \,$) is braided. This is useful because, despite these
specialized quantum groups are not quasitriangular, the braiding is enough for
applications, mainly for producing knot invariants. As an example, the action
of the $ R $--matrix on (tensor products of) Verma modules can be specialized
at odd roots of 1.
|
math
|
2,670 |
A classification of inner actions of the Dipper-Donkin quantization GL_2 on the Clifford algebra C(1,3)
|
math.QA
|
We present all inner actions on the Clifford algebra C(1,3) of the quantum
group GL_2 constructed by Dipper and Donkin.
|
math
|
2,671 |
Representations of quantum algebra U_q(u_{n,1})
|
math.QA
|
Infinite dimensional representations of the real form U_q(u_{n,1}) of the
Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of
representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of
the principal series representations are calculated in an explicit form. The
structure of reducible representations of the principal series is determined.
Irreducible representations of U_q(u_{n,1}), obtained from irreducible and
reducible principal series representations, are classified. All
*-representations in this set of irreducible representations are separated.
Unlike the classical case, the algebra U_q(u_{n,1}) has finite dimensional
irreducible *-representations.
|
math
|
2,672 |
The second cohomology of sl(m|1) with coefficients in its enveloping algebra is trivial
|
math.QA
|
Using techniques developed in a recent article by the authors, it is proved
that the 2-cohomology of the Lie superalgebra sl(m|1); m > 1, with coefficients
in its enveloping algebra is trivial. The obstacles in solving the analogous
problem for sl(3|2) are also discussed.
|
math
|
2,673 |
Rogawski's conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A
|
math.QA
|
The functors constructed by Arakawa and the author relate the representation
theory of gl_n and that of the degenerate affine Hecke algebra H_l of GL_l.
They transform the Verma modules over gl_n to the standard modules over H_l.
They transform the simple modules to the simple modules. We also prove that
they transform the Jantzen filtration on the Verma modules to that on the
standard modules. We obtain the following results for the representations of
H_l by translating the corresponding results for gl_n through the functors: (i)
the (generalized) Bernstein-Gelfand-Gelfand resolution for a certain class of
simple modules, (ii) the multiplicity formula for the composition series of the
standard modules, and (iii) its refinement concerning the Jantzen filtration on
the standard modules, which was conjectured by Rogawski.
|
math
|
2,674 |
Representations of the cyclically symmetric q-deformed algebra $so_q(3)$
|
math.QA
|
An algebra homomorphism $\psi$ from the nonstandard q-deformed (cyclically
symmetric) algebra $U_q(so_3)$ to the extension ${\hat U}_q(sl_2)$ of the Hopf
algebra $U_q(sl_2)$ is constructed. Not all irreducible representations of
$U_q(sl_2)$ can be extended to representations of ${\hat U}_q(sl_2)$. Composing
the homomorphism $\psi$ with irreducible representations of ${\hat U}_q(sl_2)$
we obtain representations of $U_q(so_3)$. Not all of these representations of
$U_q(so_3)$ are irreducible. Reducible representations of $U_q(so_3)$ are
decomposed into irreducible components. In this way we obtain all irreducible
representations of $U_q(so_3)$ when $q$ is not a root of unity. A part of these
representations turns into irreducible representations of the Lie algebra
so$_3$ when $q\to 1$. Representations of the other part have no classical
analogue. Using the homomorphism $\psi$ it is shown how to construct tensor
products of finite dimensional representations of $U_q(so_3)$. Irreducible
representations of $U_q(so_3)$ when $q$ is a root of unity are constructed.
Part of them are obtained from irreducible representations of ${\hat
U}_q(sl_2)$ by means of the homomorphism $\psi$.
|
math
|
2,675 |
Some crystal Rogers-Ramanujan type identities
|
math.QA
|
By using the Kang-Kashiwara-Misra-Miwa-Nakashima-Nakayashiki crystal base
character formula for the basic $A_2^{(1)}$-module, and the principally
specialized Weyl-Kac character formula, we obtain a Rogers-Ramanujan type
combinatorial identity for colored partitions. The difference conditions
between parts are given by the energy function of certain perfect
$A_2^{(1)}$-crystal. We also recall some other identities for this type of
colored partitions, but coming from the vertex operator constructions and with
no apparent connection to the crystal base theory.
|
math
|
2,676 |
Axioms for Weak Bialgebras
|
math.QA
|
Let A be a finite dimensional unital associative algebra over a field K,
which is also equipped with a coassociative counital coalgebra structure
(\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is
multiplicative. We do not require \Delta(1) = 1 \otimes 1 nor multiplicativity
of the counit \eps. Instead, we propose a new set of counit axioms, which are
modelled so as to guarantee that \Rep\A becomes a monoidal category with unit
object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat
A denoting the dual weak bialgebra). Under these monoidality axioms \E and
\bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are
trivial if and only if the counit \eps is multiplicative. We also propose
axioms for an antipode S such that the category \Rep\A becomes rigid. S is
uniquely determined, provided it exists. If a monoidal weak bialgebra A has an
antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra
anti-homomorphism, in which case S is also invertible. In this way we obtain a
definition of weak Hopf algebras which in Appendix A will be shown to be
equivalent to the one given independently by G. B\"ohm and K. Szlach\'anyi.
Special examples are given by the face algebras of T. Hayashi and the
generalised Kac algebras of T. Yamanouchi.
|
math
|
2,677 |
On Finite-Dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic
|
math.QA
|
Recently, important progress has been made in the study of finite-dimensional
semisimple Hopf algebras over a field of characteristic zero. Yet, very little
is known over a field of positive characteristic. In this paper we prove some
results on finite-dimensional semisimple and cosemisimple Hopf algebras A over
a field of positive characteristic, notably Kaplansky's 5th conjecture on the
order of the antipode of A. These results have already been proved over a field
of characteristic zero, so in a sense we demonstrate that it is sufficient to
consider semisimple Hopf algebras over such a field (they are also
cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for
semisimple and cosemisimple Hopf algebras over a field of positive
characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments
of deformation theory from positive to zero characteristic. The key ingredient
of the proof is the theorem that the bialgebra cohomology groups of A vanish.
|
math
|
2,678 |
The fake monster formal group
|
math.QA
|
The main result of this paper is the construction of ``good'' integral forms
for the universal enveloping algebras of the fake monster Lie algebra and the
Virasoro algebra. As an application we construct formal group laws over the
integers for these Lie algebras. We also prove a form of the no-ghost theorem
over the integers, and use this to verify an assumption used in the proof of
the modular moonshine conjectures.
|
math
|
2,679 |
Towards Drinfeld-Sokolov reduction for quantum groups
|
math.QA
|
In this paper we study the Poisson-Lie version of the Drinfeld-Sokolov
reduction defined in q-alg/9704011, q-alg/9702016. Using the bialgebra
structure related to the new Drinfeld realization of affine quantum groups we
describe reduction in terms of constraints. This realization of reduction
admits direct quantization.
As a byproduct we obtain an explicit expression for the symplectic form
associated to the twisted Heisenberg double and calculate the moment map for
the twisted dressing action. For some class of infinite-dimensional Poisson Lie
groups we also prove an analogue of the Ginzburg-Weinstein isomorphism.
|
math
|
2,680 |
A contribution of a U(1)-reducible connection to quantum invariants of links I: R-matrix and Burau representation
|
math.QA
|
We use the relation between the quantum su(2) R-matrix and the Burau
representation of the braid group in order to study the structure of the
colored Jones polynomial of links. We show that similarly to the case of a
knot, the colored Jones polynomial of a link can be presented as a formal
series in powers of q-1. The coefficients of this series are rational functions
of q^(color) whose denominators are powers of the Alexander-Conway polynomial.
|
math
|
2,681 |
Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces
|
math.QA
|
A method of constructing covariant differential calculi on a quantum
homogeneous space is devised. The function algebra X of the quantum homogeneous
space is assumed to be a left coideal of a coquasitriangular Hopf algebra H and
to contain the coefficients of any matrix over H which is the two-sided inverse
of one with entries in X. The method is based on partial derivatives. For the
quantum sphere of Podles and the quantizations of symmetric spaces due to
Noumi, Dijkhuizen and Sugitani the construction produces the subcalculi of the
standard bicovariant calculus on the quantum group.
|
math
|
2,682 |
Extended jordanian twists for Lie algebras
|
math.QA
|
Jordanian quantizations of Lie algebras are studied using the factorizable
twists. For a restricted Borel subalgebras ${\bf B}^{\vee}$ of $sl(N)$ the
explicit expressions are obtained for the twist element ${\cal F}$, universal
${\cal R}$-matrix and the corresponding canonical element ${\cal T}$. It is
shown that the twisted Hopf algebra ${\cal U}_{\cal F} ({\bf B}^{\vee})$ is
self dual. The cohomological properties of the involved Lie bialgebras are
studied to justify the existence of a contraction from the Dinfeld-Jimbo
quantization to the jordanian one. The construction of the twist is generalized
to a certain type of inhomogenious Lie algebras.
|
math
|
2,683 |
Quantum Z-algebras and representations of quantum affine algebras
|
math.QA
|
Generalizing our earlier work, we introduce the homogeneous quantum
$Z$-algebras for all quantum affine algebras $\alg$ of type one. With the new
algebras we unite previously scattered realizations of quantum affine algebras
in various cases. As a result we find a realization of $U_q(F_4^{(1)})$.
|
math
|
2,684 |
Classification of irreducible modules for the vertex operator algebra M(1)^+
|
math.QA
|
We classify the irreducible modules for the fixed point vertex operator
subebra of the rank 1 free bosonic VOA under the -1 automorphism.
|
math
|
2,685 |
Projective representation of k-Galilei group
|
math.QA
|
The projective representations of k-Galilei group G_k are found by
contracting the relevant representations of k-Poincare group. The projective
multiplier is found. It is shown that it is not possible to replace the
projective representations of G_k by vector representations of some its
extension.
|
math
|
2,686 |
Solutions of the Yang-Baxter equation and quantum sl(2)
|
math.QA
|
We construct a quantum deformation of a family of the Yang-Baxter equation
solutions naturally arising from a Lie algebra sl(2).
|
math
|
2,687 |
Zelevinsky's involution at roots of unity
|
math.QA
|
We give a combinatorial algorithm for computing Zelevinsky's involution of
the set of isomorphism classes of irreducible representations of the affine
Hecke algebra $\H_m(t)$ when $t$ is a primitive $n$th root of 1. We show that
the same map can also be interpreted in terms of aperiodic nilpotent orbits of
$\Zb/n\Zb$-graded vector spaces.
|
math
|
2,688 |
Unitarity of induced representations from coisotropic quantum groups
|
math.QA
|
We study unitarity of the induced representations from coisotropic quantum
subgroups which were introduced in math.QA/9804138. We define a real structure
on coisotropic subgroups which determines an involution on the homogeneous
space. We give general invariance properties for functionals on the homogeneous
space which are sufficient to build a unitary representation starting from the
induced one. We present the case of the one-dimensional quantum Galilei group,
where we have to use in all enerality our definition of quasi-invariant
functional.
|
math
|
2,689 |
The representation theory of free orthogonal quantum groups
|
math.QA
|
We find, for each $n\geq2$, the class of $n\times n$ compact quantum groups
whose representation theory is similar to that of $SU(2)$: this is the class of
"free analogues of $O(n)$" constructed by Van Daele and Wang.
|
math
|
2,690 |
Central extensions of classical and quantum q-Viraroso algebras
|
math.QA
|
We investigate the central extensions of the q-deformed (classical and
quantum) Virasoro algebras constructed from the elliptic quantum algebra
A_{q,p}[sl(N)_c]. After establishing the expressions of the cocycle conditions,
we solve them, both in the classical and in the quantum case (for sl(2)). We
find that the consistent central extensions are much more general that those
found previously in the literature.
|
math
|
2,691 |
A contribution of a U(1)-reducible connection to quantum invariants of links II: Links in rational homology spheres
|
math.QA
|
We extend the definition of the U(1)-reducible connection contribution to the
case of the Witten-Reshetikhin-Turaev invariant of a link in a rational
homology sphere. We prove that, similarly ot the case of a link in S^3, this
contribution is a formal power series in powers of q-1, whose coefficients are
rational functions of q^{color}, their denominators being the powers of the
Alexander-Conway polynomial. The coefficients of the polynomials in numerators
are rational numbers, the bounds on their denominators are established with the
help of the theorem proved by T. Ohtsuki in Appendix 2.
Similarly to the previously considered case of S^3, the U(1)-reducible
connection contribution determines the trivial connection contribution into the
Witten-Reshetikhin-Turaev invariant of algebraically connected links.
We derive a surgery formula for the U(1)-reducible connection contribution,
which relates it to the similar contribution into the colored Jones polynomial
of a surgery link in S^3.
|
math
|
2,692 |
Algebraic nested Bethe ansatz for the elliptic Ruijsenaars model
|
math.QA
|
The eigenvalues of the elliptic N-body Ruijsenaars operator are obtained by a
dynamical version of the algebraic nested Bethe ansatz method. We use a result
of Felder and Varchenko, who showed how to obtain the Ruijsenaars operator as
the transfer matrix of a particular representation of the elliptic quantum
group associated to gl(N).
|
math
|
2,693 |
The Ideals of Free Differential Algebras
|
math.QA
|
We consider the free ${\bf C}$-algebra ${\cal B}_q$ with $N$ generators
$\{\xi_i\}_{i = 1,...,N}$, together with a set of $N$ differential operators
$\{\partial_i\}_{i = 1,...,N}$ that act as twisted derivations on ${\cal B}_q$
according to the rule $\partial_i\xi_j = \delta_{ij} + q_{ij}\xi_j\partial_i$;
that is, $\forall x \in {\cal B}_q, \partial_i(\xi_jx) = \delta_{ij}x +
q_{ij}\xi_j\partial_i x,$ and $\partial_i{\bf C} = 0$. The suffix $q$ on ${\cal
B}_q$ stands for $\{q_{ij}\}_{i,j \in \{1,...,N\}}$ and is interpreted as a
point in parameter space, $q = \{q_{ij}\}\in {\bf C}^{N^2}$. A constant $C \in
{\cal B}_q$ is a nontrivial element with the property $\partial_iC = 0, i =
1,...,N$. To each point in parameter space there correponds a unique set of
constants and a differential complex. There are no constants when the
parameters $q_{ij}$ are in general position. We obtain some precise results
concerning the algebraic surfaces in parameter space on which constants exist.
Let ${\cal I}_q$ denote the ideal generated by the constants. We relate the
quotient algebras ${\cal B}_q' = {\cal B}_q/{\cal I}_q$ to Yang-Baxter algebras
and, in particular, to quantized Kac-Moody algebras. The differential complex
is a generalization of that of a quantized Kac-Moody algebra described in terms
of Serre generators. Integrability conditions for $q$-differential equations
are related to Hochschild cohomology. It is shown that $H^p({\cal B}_q',{\cal
B}_q') = 0$ for $p \geq 1$. The intimate relationship to generalized, quantized
Kac-Moody algebras suggests an approach to the problem of classification of
these algebras.
|
math
|
2,694 |
A method of construction of finite-dimensional triangular semisimple Hopf algebras
|
math.QA
|
The goal of this paper is to give a new method of constructing
finite-dimensional semisimple triangular Hopf algebras, including minimal ones
which are non-trivial (i.e. not group algebras). The paper shows that such Hopf
algebras are quite abundant. It also discovers an unexpected connection of such
Hopf algebras with bijective 1-cocycles on finite groups and set-theoretical
solutions of the quantum Yang-Baxter equation defined by Drinfeld.
|
math
|
2,695 |
Finite Dimensional Pointed Hopf Algebras with Abelian Coradical and Cartan matrices
|
math.QA
|
In a previous work \cite{AS2} we showed how to attach to a pointed Hopf
algebra A with coradical $\k\Gamma$, a braided strictly graded Hopf algebra R
in the category $_{\Gamma}^{\Gamma}\Cal{YD}$ of Yetter-Drinfeld modules over
$\Gamma$. In this paper, we consider a further invariant of A, namely the
subalgebra R' of R generated by the space V of primitive elements. Algebras of
this kind are known since the pioneering work of Nichols. It turns out that R'
is completely determined by the braiding c:V\otimes V \to V \otimes V. We
denote R' = B(V). We assume further that $\Gamma$ is finite abelian. Then c is
given by a matrix (b_{ij}) whose entries are roots of unity; we also suppose
that they have odd order. We introduce for these braidings the notion of
"braiding of Cartan type" and we attach a generalized Cartan matrix to a
braiding of Cartan type. We prove that B(V) is finite dimensional if its
corresponding matrix is of finite Cartan type and give sufficient conditions
for the converse statement. As a consequence, we obtain many new families of
pointed Hopf algebras. When $\Gamma$ is a direct sum of copies of a group of
prime order, the conditions hold and any matrix is of Cartan type. As a sample,
we classify all the finite dimensional pointed Hopf algebras which are
coradically graded, generated in degree one and whose coradical has odd prime
dimension p. We also characterize coradically graded pointed Hopf algebras of
order p^4, which are generated in degree one.
|
math
|
2,696 |
On p-adic propreties of the Witten-Reshetikhin-Turaev invariant
|
math.QA
|
We use the properties of the Melvin-Morton expansion of the colored Jones
polynomial in order to prove that the trivial connection contribution converges
p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant of rational homology
spheres, as it was conjectured by R. Lawrence.
|
math
|
2,697 |
From Double Hecke algebra to analysis
|
math.QA
|
We discuss q-counterparts of the Gauss integrals, a new type of Gauss-Selberg
sums at roots of unity, and q-deformations of Riemann's zeta. The paper
contains general results, one-dimensional formulas, and remarks about the
current projects involving the double affine Hecke algebras.
|
math
|
2,698 |
Super-jordanian deformation of the orthosymplectic Lie superalgebras
|
math.QA
|
The recently proposed jordanian quantization of the Lie superalgebra
$osp(1|2)$ due to the embedding $sl(2) \subset osp(1|2)$, is extended including
odd generators into the twisting element $\cal F$. This deformation is obtained
as a contraction of the quantum superalgebra ${\cal U}_{q}(osp(1|2))$.
|
math
|
2,699 |
Annihilating fields of standard modules of sl(2,C)~ and combinatorial identities
|
math.QA
|
We show that a set of local admissible fields generates a vertex algebra. For
an affine Lie algebra $\tilde\goth g$ we construct the corresponding level $k$
vertex operator algebra and we show that level $k$ highest weight $\tilde\goth
g$-modules are modules for this vertex operator algebra. We determine the set
of annihilating fields of level $k$ standard modules and we study the
corresponding loop $\tilde\goth g$ module---the set of relations that defines
standard modules. In the case when $\tilde\goth g$ is of type $A_1^{(1)}$, we
construct bases of standard modules parameterized by colored partitions and, as
a consequence, we obtain a series of Rogers-Ramanujan type combinatorial
identities.
|
math
|
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