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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 gi... | 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 alg... | 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 $B_2$
spide... | 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 con... | 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 cru... | 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
multipli... | 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 ... | 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 ... | 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 a... | 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 statem... | 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... | 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, th... | 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... | 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, ... | 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 co... | 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 construc... | 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 ... | 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
furthe... | 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... | 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 ... | 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 nondeg... | 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 surfac... | 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 dou... | 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, Nicho... | 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 quantu... | 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 space... | 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 establishe... | 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 an... | 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 ... | 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... | 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 pun... | 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 an... | 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 H... | 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... | 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 c... | 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 ... | 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 ... | 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 $ ... | 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-eq... | 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 ... | 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. orb... | 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 o... | 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 giv... | 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 ... | 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 cas... | 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... | 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
represen... | 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 space... | 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 ... | 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 ... | 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 cal... | 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 oth... | 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 r... | 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... | 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 homomor... | 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 ... | 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. Inst... | 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 fiel... | 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 in... | 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... | 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 ... | 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... | 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 t... | 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
$\Zb/n\Zb$-g... | 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... | 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 consi... | 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 funct... | 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 asso... | 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, $\fo... | 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... | 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 ... | 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 ann... | math |
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