title: Quantum differentials and the q-monopole revisited

abstract:
  The q-monopole bundle introduced previously is extended to a general
construction for quantum group bundles with non-universal differential calculi.
We show that the theory applies to several other classes of bundles as well,
including bicrossproduct quantum groups, the quantum double and combinatorial
bundles associated to covers of compact manifolds.


title: The number of primitive Vassiliev invariants up to degree 12

abstract:
  We present algorithms giving upper and lower bounds for the number of
independent primitive rational Vassiliev invariants of degree m modulo those of
degree m-1. The values have been calculated for the formerly unknown degrees m
= 10, 11, 12. Upper and lower bounds coincide, which reveals that all Vassiliev
invariants of degree smaller 13 are orientation insensitive and are coming from
representations of Lie algebras so and gl. Furthermore, a conjecture of Vogel
is falsified and it is shown that the \Lambda-module of connected trivalent
diagrams (Chinese characters) is not free.


title: Towards Deformed Chiral Algebras

abstract:
  We describe a new algebraic structure of "deformed chiral algebra" motivated
by the study of the deformed W-algebras. We use it to gain some insights into
the deformed Virasoro algebra.


title:
Classical dynamical r-matrices for Calogero-Moser systems and their
  generalizations

abstract:
  We present a derivation of the dynamical r-matrices of the Calogero-Moser
models using the Hamiltonian reduction procedure to get general formulae. We
describe the dynamical r-matrices thus found for spin Calogero-Moser models and
relativistic Ruijsenaars-Schneider models


title: Crystal Bases and Young Tableaux

abstract:
  Let B be the crystal basis of the minus part of the quantized enveloping
algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a
combinatorial description in terms of an embedding of B into the tensor product
of B and k abstract crystals B_{i_j}, j=1,2,...,k, where the longest word in
the Weyl group is s_{i_1}...s_{i_k}. We give an explicit description of the
image of this embedding for classical Lie algebras of types A, B, C, D. This
description is in terms of semi-standard Young tableaux of types A, B, C, D
defined by Kashiwara and Nakashima.


title: Quantum geometry of field extensions

abstract:
  We show that noncommutative differential forms on $k[x]$, $k$ a field, are of
the form $\Omega^1=k_\lambda[x]$ where $k_\lambda\supset k$ is a field
extension. We compute the case $C\supset R$ explicitly, where $\Omega^1$ is
2-dimensional. We study the induced quantum de Rahm complex, its cohomology and
the associated moduli space of flat connections.


title: Singular deformations of Lie algebras on an example

abstract:
  The notion of singular one-parameter deformation of a Lie algebra is
introduced. It is shown that the complex infinite-dimensional Lie algebra of
polynomial vector fields in C with trivial 1-jet at the origin has such
singular deformation.


title:
Intertwining operator algebras and vertex tensor categories for affine
  Lie algebras

abstract:
  We apply the general theory of tensor products of modules for a vertex
operator algebra developed in our papers hep-th/9309076, hep-th/9309159,
hep-th/9401119, q-alg/9505018, q-alg/9505019 and q-alg/9505020 to the case of
the Wess-Zumino-Novikov-Witten models and related models in conformal field
theory. We show that for the category of modules for a vertex operator algebra
containing a subalgebra isomorphic to a tensor product of rational vertex
operator algebras associated to affine Lie algebras, the intertwining operators
among the modules have the associativity property, the category has a natural
structure of vertex tensor category, and a number of related results hold. We
obtain, as a corollary and special case, a construction of the
previously-studied braided tensor category structure on the category of finite
direct sums of standard (integrable highest weight) modules of a fixed positive
integral level for an affine Lie algebra.


title:
The structure of quantum Lie algebras for the classical series B_l, C_l
  and D_l

abstract:
  The structure constants of quantum Lie algebras depend on a quantum
deformation parameter q and they reduce to the classical structure constants of
a Lie algebra at $q=1$. We explain the relationship between the structure
constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for
adjoint x adjoint ---> adjoint. We present a practical method for the
determination of these quantum Clebsch-Gordan coefficients and are thus able to
give explicit expressions for the structure constants of the quantum Lie
algebras associated to the classical Lie algebras B_l, C_l and D_l.
  In the quantum case also the structure constants of the Cartan subalgebra are
non-zero and we observe that they are determined in terms of the simple quantum
roots. We introduce an invariant Killing form on the quantum Lie algebras and
find that it takes values which are simple q-deformations of the classical
ones.


title: Conformal Modules

abstract:
  In this paper we study a class of modules over infinite-dimensional Lie
(super)algebras, which we call conformal modules. In particular we classify and
construct explicitly all irreducible conformal modules over the Virasoro and
the N=1 Neveu-Schwarz algebras, and over the current algebras.


title: Particles in Singular Magnetic Field

abstract:
  An algebraic formalism for description of quantum states of charged particle
with spin moving in two-dimensional space under influence of singular magnetic
field is developed in terms of graded algebras. The fundamental assumption is
that the particle is transformed into a composite system which consists
quasiparticles, quasiholes and magnetic fluxes. Such system is endowed with
generalized statistics determined by a grading group and a commutation factor
on it. Composite systems corresponding to the quantum Hall effect and the
electronic magnetotransport anomaly are described. The Fock space
representation are also given.


title: Reflection Equation Algebra of a $(h,w)$-deformed Oscillator

abstract:
  We consider the reflection equation algebra for a finite dimensional R-matrix
for the $(h,w)$-deformed Heisenberg algebra ${\cal U}_{h,w}(h(4))$. A
representation of the reflection matrix $K$ is constructed using the matrix
generators $L^{(\pm)}$ of the ${\cal U}_{h,w}(h(4))$ algebra. A series of
representations of the K-matrix then may be generated by using the coproduct
rules of the ${\cal U}_{h,w}(h(4))$ algebra. The complementary condition
necessary for combining two distinct solutions of the reflection equation
algebra yields the braiding relations between these two sets of generators.
This may be thought as a generalization of Bose-Fermi statistics to braiding
statistics, which them may be used to provide a new braided colagebraic
structure to a Hopf algebra generated by the elements of the matrix $K$. The
reflection equation algebra and the braided exchange properties are found to
depend on both deformation parameters $h$ and $w$.


title:
General Construction of Nonstandard $R_h$-matrices as Contraction Limits
  of $R_{q}$-matrices

abstract:
  A class of transformations of $R_q$-matrices is introduced such that the
$q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation
matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the
singularities, however, cancel yielding a well-defined construction. Our method
can be implemented systematically for R-matrices of all dimensions and not only
for $sl(2)$ but also for algebras of higher dimensions. Explicit constructions
are presented starting with ${\cal U}_q(sl(2))$ and ${\cal U}_q(sl(3))$, while
choosing $R_q$ for (fund. rep.)$\otimes$(arbitrary irrep.). The treatment for
the general case and various perspectives are indicated. Our method yields
nonstandard deformations along with a nonlinear map of the $h$-Borel subalgebra
on the corresponding classical Borel subalgebra. For ${\cal U}_h(sl(2))$ this
map is extended to the whole algebra and compared with another one proposed by
us previously.


title: Symmetric Jack polynomials from non-symmetric theory

abstract:
  The theory of non-symmetric Jack polynomials is developed independently of
the theory of symmetric Jack polynomials, and this theory together with the
relationship between the non-symmetric, symmetric and anti-symmetric Jack
polynomials is used to deduce the corresponding results for the symmetric Jack
polynomials.


title: Reflections on Topological Quantum Field Theory

abstract:
  (Talk presented at the XVth Workshop on Geometric Methods in Physics,
Quantizations, Deformations and Coherent States, in Bialowieza, Poland, July
1-7, 1996.) The aim of this article is to introduce some basic notions of
Topological Quantum Field Theory (TQFT) and to consider a modification of TQFT,
applicable to embedded manifolds. After an introduction based around a simple
example (Section 1) the notion of a d-dimensional TQFT is defined in
category-theoretical terms, as a certain type of functor from a category of
d-dimensional cobordisms to the category of vector spaces (Section 2). A
construction due to Turaev, an operator-valued invariant of tangles, is
discussed in Section 3. It bears a strong resemblance to 1-dimensional TQFTs,
but carries much richer structure due to the fact that the 1-dimensional
manifolds involved are embedded in a 3-dimensional space. This leads us, in
Section 4, to propose a class of TQFT-like theories, appropriate to embedded,
rather than pure, manifolds.


title: A note on quivers with symmetries

abstract:
  We show that the bases of irreducible integrable highest weight module of a
non-symmetric Kac-Moody algebra, which is associated to a quiver with a
nontrivial admissible automorphism, can be naturally identified with a set of
certain invariant Langrangian irreducible subvarieties of certain varieties
associated with the quiver defined by Nakajima. In the case of non-symmetric
affine or finite Kac-Moody algebras, the bases can be naturally identified with
a set of certain invariant Langrangian irreducible subvarieties of a particular
deformation of singularities of the moduli space of instantons over A-L-E
spaces.
  The motivation of this paper comes from string/string duality and the paper
is ended with questions and speculations related to string/string duality.


title: Characters of Demazure modules and solvable lattice models

abstract:
  We study the path realization of Demazure crystals related to solvable
lattice models in statistical mechanics. Various characters are represented in
a unified way as the sums over one dimensional configurations which we call
unrestricted, classically restricted and restricted paths. As an application
characters of Demazure modules are obtained in terms of $q$-multinomial
coefficients for several level 1 modules of classical affine algebras.


title: On BC type basic hypergeometric orthogonal polynomials

abstract:
  The five parameter family of multivariable Askey-Wilson polynomials is
studied with four parameters generically complex. The multivariable
Askey-Wilson polynomials form an orthogonal system with respect to an explicit
(in general complex) measure. A partially discrete orthogonality measure is
obtained by shifting the contour to the torus while picking up residues. A
parameter domain is given for which the partially discrete orthogonality
measure is positive. The orthogonality relations and norm evaluations for
multivariable q-Racah polynomials and multivariable big and little q-Jacobi
polynomials are proved by taking suitable limits in the orthogonality relations
for the multivariable Askey-Wilson polynomials. In particular new proofs of
several well known q-analogues of the Selberg integral are obtained.


title: Drinfeld twist for quantum $su(2)$ in the adjoint representation

abstract:
  We give a detailed description of the adjoint representation of Drinfeld's
twist element, as well as of its coproduct, for $su_{q}(2)$. We also discuss,
as applications, the computation of the universal R-matrix in this
representation and the problem of symmetrization of identical-particle states
with quantum $su(2)$ symmetry.


title:
Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson
  algebra on $\dot T^*S^1$

abstract:
  We study deformations of the standard embedding of the Lie algebra
$\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra of
functions on the cotangent bundle $T^*S^1$ (with respect to the Poisson
bracket). We consider two analogous but different problems: (a) formal
deformations of the standard embedding of $\Vect(S^1)$ into the Lie algebra of
functions on $\dot T^*S^1:=T^*S^1\setminusS^1$ which are Laurent polynomials on
fibers, and (b) polynomial deformations of the $\Vect(S^1)$ subalgebra inside
the Lie algebra of formal Laurent series on $\dot T^*S^1$.


title: Framed vertex operator algebras, codes and the moonshine module

abstract:
  For a simple vertex operator algebra whose Virasoro element is a sum of
commutative Virasoro elements of central charge 1/2, two codes are introduced
and studied. It is proved that such vertex operator algebras are rational. For
lattice vertex operator algebras and related ones, decompositions into direct
sums of irreducible modules for the product of the Virasoro algebras of central
charge 1/2 are explicitly described. As an application, the decomposition of
the moonshine vertex operator algebra is obtained for a distinguished system of
48 Virasoro algebras.


title: The prime spectrum of a quantum Bruhat cell translate

abstract:
  The prime spectra of two families of algebras, $S^w$ and $\check{S}^w$, $w\in
W,$ indexed by the Weyl group $W$ of a semisimple finitely dimensional are
studied. The algebras $S^w$ have been introduced by A.~Joseph; they are
$q$-analogues of the algebras of regular functions on $w$-translates of the
open Bruhat cell of a semisimple Lie group $G$ corresponding to the Lie algebra
$\fg$.
  We define a stratification of the spectra into components indexed by pairs
$(y_1,y_2)$ of elements of the Weyl group satisfying $y_1\leq w\leq y_2$. Each
component admits a unique minimal ideal which is explicitly described. We show
the inclusion relation of closures to be that induced by Bruhat order.


title: Knot Invariants from a Kohno-Kontsevich Integral for Braids

abstract:
  An invariant of knots is constructed from an integral for geometric braids
due to Kohno and Kontsevich. It takes values in a quotient by a certain ideal
of the algebra generated by chord diagrams over the circle.


title: The symplectic structure of the spin Calogero model

abstract:
  We compute the symplectic structure of the spin Calogero model in terms of
algebro-geometric data on the associated spectral curve.


title: Coinvariants for Yangian doubles and quantum KZ equations

abstract:
  We present a quantum version of the construction of the KZ system of
equations as a flat connection on the spaces of coinvariants of representations
of tensor products of Kac-Moody algebras. We consider here representations of a
tensor product of Yangian doubles and compute the coinvariants of a deformation
of the subalgebra generated by the regular functions of a rational curve with
marked points. We observe that Drinfeld's quantum Casimir element can be viewed
as a deformation of the zero-mode of the Sugawara tensor in the Yangian double.
These ingredients serve to define a compatible system of difference equations,
which we identify with the quantum KZ equations introduced by I. Frenkel and N.
Reshetikhin.


title: q-Uncertainty Relations

abstract:
  As one would anticipate from the realization of the q-commutators by
difference operators, the states of maximum localization are smeared delta
functions depending on q.


title: Crystals for Demazure Modules of Classical Affine Lie Algebras

abstract:
  We study, in the path realization, crystals for Demazure modules of affine
Lie algebras of types $A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n,
A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$. We find a special sequence of
affine Weyl group elements for the selected perfect crystal, and show if the
highest weight is $l\La_0$, the Demazure crystal has a remarkably simple
structure.


title: On braided Poisson and quantum inhomogeneous groups

abstract:
  The well known incompatibility between inhomogeneous quantum groups and the
standard q-deformation is shown to disappear (at least in certain cases) when
admitting the quantum group to be braided. Braided quantum ISO(p,N-p)
containing SO_q(p,N-p) with |q|=1 are constructed for N=2p, 2p+1, 2p+2. Their
Poisson analogues (obtained first) are presented as an introduction to the
quantum case.


title: Connes' distance function on one-dimensional lattices

abstract:
  We show that there is an operator with a simple geometric significance which
yields the ordinary geometry of a linear equidistant lattice via Connes'
distance function.


title: Nonabelian Integrable Systems, Quasideterminants, and Marchenko Lemma

abstract:
  We find explicit (multisoliton) solutions for nonabelian integrable systems
such as periodic Toda field equations, Langmuir equations, and Schrodinger
equations for functions with values in any associative algebra. The solution
for nonabelian Toda field equations for root systems of types A, B, C was
expressed by the authors in a previous paper (q-alg/9701008) using
quasideterminants introduced by the last two authors. To find multisoliton
solutions of periodic Toda equations and other nonabelian systems we use a
combination of these ideas with important lemmas which are due to Marchenko.


title: Automorphisms of the Weyl algebra and bispectral operators

abstract:
  In our previous paper q-alg/9605011 we proposed several algebraic methods for
constructing new solutions to the bispectral problem. In the present note the
corresponding eigenfunctions are explicitly constructed as multiple Laplace
integrals.


title: Dynamical $r$-matrices for Hitchin's systems on Schottky curves

abstract:
  We express Hitchin's systems on curves in Schottky parametrization, and
construct dynamical $r$-matrices attached to them.


title: Bargmann representations for deformed harmonic oscillators

abstract:
  Generalizing the case of the usual harmonic oscillator, we look for Bargmann
representations corresponding to deformed harmonic oscillators. Deformed
harmonic oscillator algebras are generated by four operators $a, a^\dagger, N$
and the unity 1 such as $[a,N] = a, [a^\dagger,N] = -a^\dagger$, $a^\dagger a =
\psi(N)$ and $aa^\dagger =\psi(N+1)$. We discuss the conditions of existence of
a scalar product expressed with a true integral on the space spanned by the
eigenstates of $a$ (or $a^\dagger$). We give various examples, in particular we
consider functions $\psi$ that are linear combinations of $q^N$, $q^{-N}$ and
unity and that correspond to q-oscillators with Fock-representations or with
non-Fock-representations.


abstract:
  We study Hopf-Galois extensions with central invariants for a finite
dimensional Hopf algebra. We collect general facts about them and discuss some
examples arising in the study of restricted Lie algebras and quantum groups at
roots of unity. Our focus is on representation theory and its special feature
in this situation, restriction of the central character to the subalgebra of
invariants.


title: Algebraic geometry of Hopf-Galois extensions

abstract:
  We continue the study of Hopf-Galois extensions with central invariants for a
finite dimensional Hopf algebra. We concentrate on the geometrical side on the
subject. We understand how to localize Hopf-Galois extensions and to paste them
from local datum.


title:
Deformed harmonic oscillators : coherent states and Bargmann
  representations


title:
Some comments on $q$-deformed oscillators and $q$-deformed $su(2)$
  algebras

abstract:
  The various relations between $q$-deformed oscillators algebras and the
$q$-deformed $su(2)$ algebras are discussed. In particular, we exhibit the
similarity of the $q$-deformed $su(2)$ algebra obtained from $q$-oscillators
via Schwinger construction and those obtained from $q$-Holstein-Primakoff
transformation and show how the relation between $su_{\sqrt{q}}(2)$ and Hong
Yan $q$-oscillator can be regarded as an special case of In\"{o}u\"{e}- Wigner
contraction. This latter observation and the imposition of positive norm
requirement suggest that Hong-Yan $q$-oscillator algebra is different from the
usual $su_{\sqrt{q}}(2)$ algebra, contrary to current belief in the literature.


title: Bicrossproduct structure of the null-plane quantum Poincare algebra

abstract:
  A nonlinear change of basis allows to show that the non-standard quantum
deformation of the (3+1) Poincare algebra has a bicrossproduct structure.
Quantum universal R-matrix, Pauli-Lubanski and mass operators are presented in
the new basis.


title: Explicit Decompositions of Weyl Reflections in Affine Lie Algebras

abstract:
  In this paper explicit decompositions are provided of the Weyl reflections in
affine Lie algebras, in terms of fundamental Weyl reflections.


title: On set-theoretical solutions of the quantum Yang-Baxter equation

abstract:
  Recently V.Drinfeld formulated a number of problems in quantum group theory.
In particular, he suggested to consider ``set-theoretical'' solutions of the
quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the
set $X\times X$, where $X$ is a fixed finite set. In this note we study such
solutions, which satisfy the unitarity and the crossing symmetry conditions --
natural conditions arising in physical applications. More specifically, we
consider ``linear'' solutions: the set $X$ is an abelian group, and the map $R$
is an automorphism of $X\times X$. We show that in this case, solutions are in
1-1 correspondence with pairs $a,b\in \End X$, such that $b$ is invertible and
$bab^{-1}=\frac{a}{a+1}$. Later we consider ``affine'' solutions ($R$ is an
automorphism of $X\times X$ as a principal homogeneous space), and show that
they have a similar classification. The fact that these classifications are so
nice leads us to think that there should be some interesting structure hidden
behind this problem.


title: A Littlewood-Richardson Rule for factorial Schur functions

abstract:
  We give a combinatorial rule for calculating the coefficients in the
expansion of a product of two factorial Schur functions. It is a special case
of a more general rule which also gives the coefficients in the expansion of a
skew factorial Schur function. Multiplication rules for the Capelli operators
and quantum immanants are also given.


title: On the Hopf algebra structure of perturbative quantum field theories

abstract:
  We show that the process of renormalization encapsules a Hopf algebra
structure in a natural manner. This sheds light on the recently proposed
connection between knots and renormalization theory.


title:
Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and
  Standard Ordering with Differential Operator Representation

abstract:
  In this paper we construct homogeneous star products of Weyl type on every
cotangent bundle $T^*Q$ by means of the Fedosov procedure using a symplectic
torsion-free connection on $T^*Q$ homogeneous of degree zero with respect to
the Liouville vector field. By a fibrewise equivalence transformation we
construct a homogeneous Fedosov star product of standard ordered type
equivalent to the homogeneous Fedosov star product of Weyl type.
Representations for both star product algebras by differential operators on
functions on $Q$ are constructed leading in the case of the standard ordered
product to the usual standard ordering prescription for smooth complex-valued
functions on $T^*Q$ polynomial in the momenta (where an arbitrary fixed
torsion-free connection $\nabla_0$ on $Q$ is used). Motivated by the flat case
$T^* R^n$ another homogeneous star product of Weyl type corresponding to the
Weyl ordering prescription is constructed. The example of the cotangent bundle
of an arbitrary Lie group is explicitly computed and the star product given by
Gutt is rederived in our approach.


title: Double quantization on the coadjoint representation of sl(n)

abstract:
  For $\g=sl(n)$ we construct a two parametric $U_h(\g)$-invariant family of
algebras, $(S\g)_{t,h}$, which defines a quantization of the function algebra
$S\g$ on the coadjoint representation and in the parameter $t$ gives a
quantization of the Lie bracket. The family induces a two parametric
deformation of the function algebra of any maximal orbit which is a
quantization of the Kirillov-Kostant-Souriau bracket in the parameter $t$. In
addition we construct a quantum de Rham complex on $\g^*$.


title:
Classification of Bicovariant Differential Calculi on the Quantum Groups
  $SL_q(n+1)$ and $Sp_q(2n)$

abstract:
  For transcendental values of $q$ all bicovariant first order differential
calculi on the coordinate Hopf algebras of the quantum groups $SL_q(n+1)$ and
$Sp_q(2n)$ are classified. It is shown that the irreducible bicovariant first
order calculi are determined by an irreducible corepresentation of the quantum
group and a complex number $\zeta$ such that $\zeta^{n+1}=1$ for $SL_q(n+1)$
and $\zeta^2=1$ for $Sp_q(2n)$. Any bicovariant calculus is inner and its
quantum Lie algebra is generated by a central element. The main technical
ingredient is a result of the Hopf algebra $R(G_q)^0$ for arbitrary simple Lie
algebras.


title:
The q-Deformed Oscillator Representations and Their Coherent States of
  the su(1,1) Algebra

abstract:
  We present various oscillator representations of the q-deformed su(1,1)
algebra such as the Holstein-Primakoff, the Dyson, the Fock-Bargmann, the
anyonic, and the parabose oscillator representations and discuss their coherent
states with the resolution of unity.


title: From quantum to elliptic algebras

abstract:
  It is shown that the elliptic algebra ${\cal A}_{q,p}(\hat{sl}(2)_c)$ at the
critical level c=-2 has a multidimensional center containing some trace-like
operators t(z). A family of Poisson structures indexed by a non-negative
integer and containing the q-deformed Virasoro algebra is constructed on this
center. We show also that t(z) close an exchange algebra when p^m=q^{c+2} for m
integer, they commute when in addition p=q^{2k} for k integer non-zero, and
they belong to the center of ${\cal A}_{q,p}(\hat{sl}(2)_c)$ when k is odd. The
Poisson structures obtained for t(z) in these classical limits contain the
q-deformed Virasoro algebra, characterizing the structures at generic values of
p, q and m as new ${\cal W}_{q,p}(sl(2))$ algebras.


title: On quantum topology, hypergraphs and flag vectors

abstract:
  Each rule $f$ that assigns a vector $f(G)$ to an $(n+1)$-graph $G$ determines
a class (or property) of $n$-manifold invariants. An invariant $v=v(M)$ is in
this class if, for any triangulated manifold $|G|=M$, one has that $v(M)$ is a
linear function of $f(G)$. This paper defines a flag vector $f(G)$ for
$i$-graphs, whose associated invariants might be quantum, and which is of
interest in its own right. The definition (via the concept of shelling, and a
`disjoint pair of optional cells' rule for the link) seems to apply to any
finite combinatorial object, and so to any compact topological object that can
be triangulated. It also applies to finite groups.


title: The Alexander polynomial and finite type 3-manifold invariants

abstract:
  This is a revised version (replacing an older one) with typos fixed and the
introduction expanded.


title:
A Natural Basis for Spinor and Vector Fields on the Noncommutative
  sphere

abstract:
  The product of two Heisenberg-Weil algebras contains the Jordan-Schwinger
representation of su(2). This Algebra is quotiented by the square-root of the
Casimir to produce a non-associative algebra denoted by $\Psi$. This algebra
may be viewed as the right-module over one of its associative subalgebras which
corresponds to the algebra of scalar fields on the noncommutative sphere. It is
now possible to interpret other subspaces as the space of spinor or vector
fields on the noncommutative sphere. A natural basis of $\Psi$ is given which
may be interpreted as the deformed entries in the rotation matrices of SU(2).


title: Diagonal Crossed Products by Duals of Quasi-Quantum Groups

abstract:
  Let \G be a (weak) quasi-Hopf algebra. Using a two-sided \G-coaction on an
algebra \M, we construct what we call the diagonal crossed product as a new
associative algebra structure on \M\otimes \dG, where \dG is the dual of \G.
This construction is largely motivated by the special case \M = \G, for which
we obtain an explicit definition of the quantum double \D(\G) for quasi-Hopf
algebras. Applications of our formalism include the field algebra construction
of Mack and Schomerus as well as the formulation of Hopf Spin chains or lattice
current algebras based on truncated quantum groups at roots of unity. A
complete proof that \D(\G) is even a (weak) quasi-triangular quasi-Hopf algebra
will be given in a separate paper.


title: Diffeomorphism-Invariant Spin Network States

abstract:
  We extend the theory of diffeomorphism-invariant spin network states from the
real-analytic category to the smooth category. Suppose that G is a compact
connected semisimple Lie group and P -> M is a smooth principal G-bundle. A
`cylinder function' on the space of smooth connections on P is a continuous
complex function of the holonomies along finitely many piecewise smoothly
immersed curves in M. We construct diffeomorphism-invariant functionals on the
space of cylinder functions from `spin networks': graphs in M with edges
labeled by representations of G and vertices labeled by intertwining operators.
Using the `group averaging' technique of Ashtekar, Marolf, Mourao and Thiemann,
we equip the space spanned by these `diffeomorphism-invariant spin network
states' with a natural inner product.


title: Deformations of W-algebras associated to simple Lie algebras

abstract:
  Deformed $\W$--algebra $\W_{q,t}(\g)$ associated to an arbitrary simple Lie
algebra $\g$ is defined together with its free field realizations and the
screening operators. Explicit formulas are given for generators of
$\W_{q,t}(\g)$ when $\g$ is of classical type. These formulas exhibit a deep
connection between $\W_{q,t}(\g)$ and the analytic Bethe Ansatz in integrable
models associated to quantum affine algebras $U_q(\G)$ and $U_t(\GL)$. The
scaling limit of $\W_{q,t}(\g)$ is closely related to affine Toda field
theories.


title: On Matrix Quantum Groups of type $A_n$

abstract:
  Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a
matrix Hopf algebra $H_R$, which are called function rings on the matrix
quantum semi-group and matrix quantum groups associated to $R$. We show that
for an even Hecke symmetry, the rational representations of the corresponding
quantum group are absolutely reducible and that the fusion coefficients of
simple representations depend only on the rank of the Hecke symmetry. Further
we compute the quantum rank of simple representations. We also show that the
quantum semi-group is ``Zariski'' dense in the quantum group. Finally we give a
formula for the integral.


title:
W{1+ \infty} algebra, W_3 algebra, and Friedan-Martinec-Shenker
  bosonization

abstract:
  We show that the vertex algebra W{1+ \infty} with central charge -1 is
isomorphic to a tensor product of the simple W_3 algebra with central charge -2
and a Heisenberg vertex algebra generated by a free bosonic field. We construct
a family of irreducible modules of the W_3 algebra with central charge -2 in
terms of free fields and calculate the full character formulas of these modules
with respect to the full Cartan subalgebra of the W_3 algebra.


title:
String field theory and quantum groups. I. Quantum group structures in
  geometric quantization of a self-interacting string field

abstract:
  The paper is devoted to a description of quantum group structures in the
geometric quantization of a self-interacting string field, which appear under a
transition from a tree-level of the theory to the account of loop effects in
nonperturbative quantum field theory of strings.


title: Coalgebra Extensions and Algebra Coextensions of Galois Type

abstract:
  The notion of a coalgebra-Galois extension is defined as a natural
generalisation of a Hopf-Galois extension. It is shown that any
coalgebra-Galois extension induces a unique entwining map $\psi$ compatible
with the right coaction. For the dual notion of an algebra-Galois coextension
it is also proven that there always exists a unique entwining structure
compatible with the right action.


title: Applications of braided endomorphisms from conformal inclusions

abstract:
  We give three applications of general theory about braided endomorphisms from
conformal inclusions developed previously by us. The first is an example of
subfactors associated with conformal inclusion whose dual fusion ring is
non-commutative. In the second application we show that the Kac-Wakimoto
hypothesis about certain relations between branching rules and S-matrices,
which has existed for almost a decade, is not true in at least three examples.
Finally we show that the fusion rings of subfactors associated with the
conformal inclusions $SU(n)_{(n+2)}\subset SU(n(n+1)/2)$ and $SU(n+2)_n\subset
SU((n+1)(n+2)/2)$ are canonically isomorphic using a version of level-rank
duality.


title: Star Products for integrable Poisson Structures on R^3

abstract:
  We prove the existence of a deformation quantization for integrable Poisson
structures on R^3 and give a generalization for a special class of three
dimensional manifolds.


title: Higher Order Differential Calculus on $SL_q(N)$

abstract:
  Let $\Gamma$ be an $N^2$-dimensional bicovariant first order differential
calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to
construct a differential Z-graded Hopf algebra $\Gamma^\wedge$ which contains
$\Gamma$ as its first order part. Let $q$ be a transcendental complex number.
For $N>2$ these three Z-graded Hopf algebras coincide. For Woronowicz' external
algebra we calculate the dimensions of the spaces of left-invariant and
bi-invariant $k$-forms. In this case each bi-invariant form is closed. In case
of $4D_\pm$ calculi on $SL_q(2)$ the universal calculus is strictly larger than
the other two calculi. In particular, the bi-invariant 1-form is not closed.


title: A Note on First Order Differential Calculus on Quantum Principal Bundles

abstract:
  The relationship between the exactness of a first order differential calculus
on a comodule algebra $P$ and the Galois property of $P$ is investigated.


title:
Solutions of the quantum dynamical Yang-Baxter equation and dynamical
  quantum groups

abstract:
  The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization
of the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and
Felder. The QDYB equation and its quasiclassical analogue (the classical
dynamical Yang-Baxter equation) arise in several areas of mathematics and
mathematical physics (conformal field theory, integrable systems,
representation theory). The most interesting solution of the QDYB equation is
the elliptic solution, discovered by Felder. In this paper, we prove the first
classification results for solutions of the QDYB equation. These results are
parallel to the classification of solutions of the classical dynamical
Yang-Baxter equation, obtained in our previous paper q-alg/9703040. All
solutions we found can be obtained from Felder's elliptic solution by a
limiting process and gauge transformations. Fifteen years ago the quantum
Yang-Baxter equation gave rise to the theory of quantum groups. Namely, it
turned out that the language of quantum groups (Hopf algebras) is the adequate
algebraic language to talk about solutions of the quantum Yang-Baxter equation.
In this paper we propose a similar language, originating from Felder's ideas,
which we found to be adequate for the dynamical Yang-Baxter equation. This is
the language of dynamical quantum groups (or $\h$-Hopf algebroids), which is
the quantum counterpart of the language of dynamical Poisson groupoids,
introduced in our previous paper q-alg/9703040.


title: Classification of irreducible modules of W_3 algebra with c = -2

abstract:
  We construct irreducible modules V_{\alpha}, \alpha \in \C over W_3 algebra
with c = -2 in terms of a free bosonic field. We prove that these modules
exhaust all the irreducible modules of W_3 algebra with c = -2. Highest weights
of modules V_{\alpha}, \alpha \in \C with respect to the full (two-dimensional)
Cartan subalgebra of W_3 algebra are (\alpha(\alpha -1)/2, \alpha(\alpha
-1)(2\alpha -1)/6). They are parametrized by points (t, w) on a rational curve
w^2 - t^2 (8t + 1)/9 = 0. Irreducible modules of vertex algebra W_{1+\infty}
with c = -1 are also classified.


title:
Drinfel'd Twist and q-Deforming Maps for Lie Group Covariant Heisenberg
  Algebras

abstract:
  Any deformation of a Weyl or Clifford algebra can be realized through a
change of generators in the undeformed algebra. q-Deformations of Weyl or
Clifford algebrae that were covariant under the action of a simple Lie algebra
g are characterized by their being covariant under the action of the quantum
group $U_q g$. We present a systematic procedure for determining all possible
corresponding changes of generators, together with the corresponding
realizations of the $U_q g$-action. The intriguing relation between
g-invariants and $U_q g$-invariants suggests that these changes of generators
might be employed to simplify the dynamics of some g-covariant quantum physical
systems.


title: Symmetries in the fourth Painleve equation and Okamoto polynomials

abstract:
  We propose a new representation of the fourth Painlev\'e equation in which
the $A^{(1)}_2$-symmetries become clearly visible. By means of this
representation, we clarify the internal relation between the fourth Painlev\'e
equation and the modified KP hierarchy. We obtain in particular a complete
description of the rational solutions of the fourth Painlev\'e equation in
terms of Schur functions. This implies that the so-called Okamoto polynomials,
which arise from the $\tau$-functions for rational solutions, are in fact
expressible by the 3-reduced Schur functions.


title: Applications of the lantern identity

abstract:
  The purpose of this note is to unify the role of the lantern identity in the
proof of several finiteness theorems. In particular, we show that for every
nonnegative integer m, the vector space (over the rationals) of type m (resp.
T-type m) invariants of integral homology 3-spheres are finite dimensional.
These results have already been obtained by [Oh] and [GL2] respectively; our
derivation however is simpler, conceptual and relates to several other
applications of the lantern identity.


title: Quantum groupoids and deformation quantization

abstract:
  The purpose of this Note is to unify quantum groups and star-products under a
general umbrella: quantum groupoids. It is shown that a quantum groupoid
naturally gives rise to a Lie bialgebroid as a classical limit. The converse
question, i.e., the quantization problem, is posed. In particular, any regular
triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a
Poisson manifold, its quantization is equivalent to a star-product.


title: simplicial cohomology of orbifolds

abstract:
  For any orbifold M, we explicitly construct a simplicial complex S(M) from a
given triangulation of the `coarse' underlying space together with the local
isotropy groups of M. We prove that, for any local system on M, this complex
S(M) has the same cohomology as M. The use of S(M) in explicit calculations is
illustrated in the example of the `teardrop' orbifold.


title: Anyonic Lie Algebras

abstract:
  We introduce anyonic Lie algebras in terms of structure constants. We provide
the simplest examples and formulate some open problems.


title: Doubles of Quasi-Quantum Groups

abstract:
  Drinfeld showed that any finite dimensional Hopf algebra \G extends to a
quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the
construction of a so--called diagonal crossed product developed by the authors,
we generalize this result to the case of quasi--Hopf algebras \G. As for
ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G)
is isomorphic to the tensor product of \G and its dual \dG. We give explicit
formulas for the product, the coproduct, the R--matrix and the antipode on
\D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular
quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra
containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is
not a subalgebra of \D(\G) unless the coproduct on \G is strictly
coassociative. It is shown that the category of finite dimensional
representations of \D(\G) coincides with what has been called the double
category of \G--modules by S. Majid [M2]. Thus our construction gives a
concrete realization of Majid's abstract definition of quasi--quantum doubles
in terms of a Tannaka--Krein--like reconstruction procedure. The whole
construction is shown to generalize to weak quasi--Hopf algebras with \D(\G)
now being linearly isomorphic to a subspace of \dG \otimes \G.


title: Deformed Yangians and Integrable Models

abstract:
  Twisted Hopf algebra $sl_\xi(2)$ gives rise to a deformation of the Yangian
${\cal Y}(sl(2))$. The corresponding deformations of the integrable XXX-spin
chain and the Gaudin model are discussed.


title: Quantum Lorentz and braided Poincare groups

abstract:
  Quantum Lorentz groups H admitting quantum Minkowski space V are selected.
Natural structure of a quantum space G = V x H is introduced, defining a
quantum group structure on G only for triangular H (q=1). We show that it
defines a braided quantum group structure on G for |q|=1.


title: On Quantum Orbit Method

abstract:
  A version of quantum orbit method is presented for real forms of equal rank
of quantum complex simple groups. A quantum moment map is constructed, based on
the canonical isomorphism between a quantum Heisenberg algebra and an algebra
of functions on a family of quantum G-spaces. For the series $A$, we construct
some irreducible $*$-representations of $U_q{\frak g}$ which correspond to the
semi-simple dressing orbits of minimal dimension in the dual Poisson Lie group.
It is shown that some complimentary series representations correspond to some
quantum 'tunnel' G-spaces which do not have a quasi-classical analog.


title:
Topics in isotopic pairs and their representations. III. Bunches of Lie
  algebras and modified classical Yang-Baxter equation

abstract:
  This third part of the series is a brief comment to certain aspects of the
theory of classical $r$-matrix and bihamiltonian formalism, which motivations
lie in constructions of the previous two parts.


title: Topics in hidden symmetries. VI

abstract:
  In this note devoted to some aspects of the inverse problem of representation
theory the attention is concentrated on the interrelations between various
algebraic structures (algebras with operators) unraveled by different solutions
of the same inverse problem. The Lie algebras and Jordan triple systems with
operators are considered.


title: The Casson-Walker-Lescop Invariant as a Quantum 3-manifold Invariant

abstract:
  We give a direct computational proof that the degree 1 part of the
Le-Murakami-Ohtsuki invariant of a closed oriented 3-manifold M is determined
by the Casson-Walker-Lescop invariant. Moreover, if the first Betti number of M
is equal to 2, the Le-Murakami-Ohtsuki invariant is determined by the Lescop
generalization of the Casson-Walker invariant.


title: Quantization of Equivariant Vector Bundles

abstract:
  The quantization of vector bundles is defined. Examples are constructed for
the well controlled case of equivariant vector bundles over compact coadjoint
orbits. (Coadjoint orbits are symplectic spaces with a transitive, semisimple
symmetry group.) In preparation for the main result, the quantization of
coadjoint orbits is discussed in detail.
  This subject should not be confused with the quantization of the total space
of a vector bundle such as the cotangent bundle.


title:
Explicit Hopf-Galois description of $SL_{e^{2i\pi/3}}$-induced Frobenius
  homomorphisms

abstract:
  The exact sequence of ``coordinate-ring'' Hopf algebras A(SL(2,C)) ->
A(SL_q(2)) -> A(F) determined by the Frobenius map Fr, and the same way
obtained exact sequence of (quantum) Borel subgroups, are studied when q is a
cubic root of unity. An A(SL(2,C))-linear splitting of A(SL_q(2)) making
A(SL(2,C)) a direct summand of A(SL_q(2)) is constructed and used to prove that
A(SL_q(2)) is a faithfully flat A(F)-Galois extension of A(SL(2,C)). A cocycle
and coaction determining the bicrossed-product structure of the
upper-triangular (Borel) quantum subgroup of A(SL_q(2)) are computed
explicitly.


title: Bilinear Identity for q-Hypergeometric Integrals

abstract:
  We describe a bilinear identity satisfied by certain multidimensional
q-hypergeometric integrals. The identity can be considered as a deformation of
the Riemann bilinear relation for the twisted de Rham (co)homologies.
  The identity also gives an explicit formula for scalar products of solutions
of the qKZ and the dual qKZ equations, therefore providing a deformation of
Gaudin-Korepin's formula for norms of the Bethe vectors.


title:
On the nonhamiltonian interaction of two rotators. II. Unraveling the
  algebraic structure

abstract:
  This note being is devoted to the unraveling of the algebraic structure,
which governs the quadratic nonhamiltonian interaction of two rotators
described in the first part. It should be considered in the context of a
general ideology of the inverse problem of representation theory (see
mp_arc/96-477). The resulted objects are formalized as Lie algebras with
operators (Lie algebras with additional unary operations).


title:
A q-deformation of the parastatistics and an alternative to the
  Chevalley description of $U_q[osp(2n+1/2m)]$

abstract:
  The paper contains essentially two new results. Physically, a deformation of
the parastatistics in a sense of quantum groups is carried out. Mathematically,
an alternative to the Chevalley description of the quantum orthosymplectic
superalgebra U_q[osp(2n+1/2m)] in terms of $m$ pairs of deformed parabosons and
$n$ pairs of deformed parafermions is outlined.


title:
Highest weight irreducible representations of the quantum algebra
  $U_h(A_\infty)$

abstract:
  A class of highest weight irreducible representations of the algebra
$U_h(A_\infty)$, the quantum analogue of the completion and central extension
$A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably
larger than the known so far representations. Within each module a basis is
introduced and the transformation relations of the basis under the action of
the Chevalley generators are explicitly written. The verification of the
quantum algebra relations to be satisfied is shown to reduce to a set of
nontrivial $q$-number identities. All our representations are restricted in the
terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. 140,
399-414 (1991)).


title: The Jordanian deformation of su(2) and Clebsch-Gordan coefficients

abstract:
  Representation theory for the Jordanian quantum algebra U=U_h(sl(2)) is
developed using a nonlinear relation between its generators and those of sl(2).
Closed form expressions are given for the action of the generators of U on the
basis vectors of finite dimensional irreducible representations. In the tensor
product of two such representations, a new basis is constructed on which the
generators of U have a simple action. Using this basis, a general formula is
obtained for the Clebsch-Gordan coefficients of U. It is shown that the
Clebsch-Gordan matrix is essentially the product of a triangular matrix with an
su(2) Clebsch-Gordan matrix. Using this fact, some remarkable properties of
these Clebsch-Gordan coefficients are derived.


title: $\Gamma$-Conformal Algebras

abstract:
  $\Gamma$-conformal algebra is an axiomatic description of the operator
product expansion of chiral fields with simple poles at finitely many points.
We classify these algebras and their representations in terms of Lie algebras
and their representations with an action of the group $\Gamma$. To every
$\Gamma$-conformal algebra and a character of $\Gamma$ we associate a Lie
algebra generated by fields with the OPE with simple poles. Examples include
twisted affine Kac-Moody algebras, the sin algebra (which is a
``$\Gamma$-conformal'' analodue of the general linear algebra) and its
analogues, the algebra of pseudodifferential operators on the circle, etc.


title: Noncommutative Geometry of the h-deformed Quantum Plane

abstract:
  The $h$-deformed quantum plane is a counterpart of the $q$-deformed one in
the set of quantum planes which are covariant under those quantum deformations
of GL(2) which admit a central determinant. We have investigated the
noncommutative geometry of the $h$-deformed quantum plane. There is a
2-parameter family of torsion-free linear connections, a 1-parameter sub-family
of which are compatible with a skew-symmetric non-degenerate bilinear map. The
skew-symmetric map resembles a symplectic 2-form and induces a metric. It is
also shown that the extended $h$-deformed quantum plane is a noncommutative
version of the Poincar\'e half-plane, a surface of constant negative Gaussian
curvature.


title: The idea of locality

abstract:
  This is a review of recent results on conformal (super)algebras. It may be
viewed as an amplification of my Wigner medal acceptance speech (given in July
1996 in Goslar, Germany) reproduced in the introduction.


title:
Quantum two-photon algebra from non-standard U_z(sl(2,R)) and a discrete
  time Schr\"odinger equation

abstract:
  The non-standard quantum deformation of the (trivially) extended sl(2,R)
algebra is used to construct a new quantum deformation of the two-photon
algebra h_6 and its associated quantum universal R-matrix. A deformed one-boson
representation for this algebra is deduced and applied to construct a first
order deformation of the differential equation that generates the two-photon
algebra eigenstates in Quantum Optics. On the other hand, the isomorphism
between h_6 and the (1+1) Schr\"odinger algebra leads to a new quantum
deformation for the latter for which a differential-difference realization is
presented. From it, a time discretization of the heat-Schr\"odinger equation is
obtained and the quantum Schr\"odinger generators are shown to be symmetry
operators.


title:
Domino tableaux, Schutzenberger involution, and the symmetric group
  action

abstract:
  We define an action of the symmetric group on the set of domino tableaux, and
prove that the number of domino tableaux of a given weight does not depend on
the permutation of components of the last. A bijective proof of the well-known
result due to J. Stembridge that the number of self-evacuating tableaux of a
given shape is equal to that of domino tableaux of the same shape is given.


title:
Asymptotics of Jack polynomials as the number of variables goes to
  infinity

abstract:
  In this paper we study the asymptotic behavior of the Jack rational functions
as the number of variables grows to infinity. Our results generalize the
results of A. Vershik and S. Kerov obtained in the Schur function case
(theta=1). For theta=1/2,2 our results describe approximation of the spherical
functions of the infinite-dimensional symmetric spaces $U(\infty)/O(\infty)$
and $U(2\infty)/Sp(\infty)$ by the spherical functions of the corresponding
finite-dimensional symmetric spaces.

categories: q-alg cond-mat hep-th math.QA nlin.SI solv-int


title: Noncommutativity and Discrete Physics

abstract:
  The purpose of this paper is to present an introduction to a point of view
for discrete foundations of physics. In taking a discrete stance, we find that
the initial expression of physical theory must occur in a context of
noncommutative algebra and noncommutative vector analysis. In this way the
formalism of quantum mechanics occurs first, but not necessarily with the usual
interpretations. The basis for this work is a non-commutative discrete calculus
and the observation that it takes one tick of the discrete clock to measure
momentum.


title: An Elliptic Algebra $U_{q,p}(\hat{sl_2})$ and the Fusion RSOS Model

abstract:
  We introduce an elliptic algebra $U_{q,p}(\hat{sl_2})$ with $p=q^{2r} (r\in
\R_{>0})$ and present its free boson representation at generic level $k$. We
show that this algebra governs a structure of the space of states in the
$k-$fusion RSOS model specified by a pair of positive integers $(r,k)$, or
equivalently a $q-$deformation of the coset conformal field theory
$SU(2)_k\times SU(2)_{r-k-2}/SU(2)_{r-2}$. Extending the work by Lukyanov and
Pugai corresponding to the case $k=1$, we gives a full set of screening
operators for $k>1$. The algebra $U_{q,p}(\hat{sl_2})$ has two interesting
degeneration limits, $p\to 0$ and $p\to 1$. The former limit yields the quantum
affine algebra $U_{q}(\hat{sl_2})$ whereas the latter yields the algebra ${\cal
A}_{\hbar,\eta}(\hat{sl_2})$, the scaling limit of the elliptic algebra ${\cal
A}_{q,p}(\hat{sl_2})$. Using this correspondence, we also obtain the highest
component of two types of vertex operators which can be regarded as
$q-$deformations of the primary fields in the coset conformal field theory.


title:
Quantum currents realization of the elliptic quantum groups
  $E_{\tau,\eta}({\frak{sl}}_{2})$

abstract:
  We review the construction by G. Felder and the author of the realization of
the elliptic quantum groups of sl(2)-type by quantum currents.


title: The reduced Birman-Wenzl algebra of Coxeter type B

abstract:
  We introduce a reduced form of a Birman-Murakami-Wenzl Algebra associated to
the braid group of Coxeter type B and investigate its semisimplicity, Bratteli
diagram and Markov trace. Applications in knot theory and physics are outlined.


title: Clifford Hopf-gebra and Bi-universal Hopf-gebra

abstract:
  We consider a pair of independent scalar products, one scalar product on
vectors, and another independent scalar product on dual space of co-vectors.
The Clifford co-product of multivectors is calculated from the dual Clifford
algebra. With respect to this co-product unit is not group-like and vectors are
not primitive. The Clifford product and the Clifford co-product fits to the
bi-gebra with respect to the family of the (pre)-braids. The Clifford bi-gebra
is in a braided category iff at least one of these scalar products vanish.


title: On Vassiliev Invariants for Links in Handlebodies

abstract:
  The notion of Vassiliev algebra in case of hanlebodies is developed. The
analogues of the results of John Baez for links in handlebodies are proved.
That means that there exists a one-to-one correspondence between the special
class of finite type invariants of links in hanlebodies and the homogeneous
Markov traces on Vassiliev algebras. This approach uses the singular braid
monoid and braid group in a handlebody and the generalizations of the theorem
of J. Alexander and the theorem of A. A. Markov for singular links and braids
and the relative version of Markov's theorem.


title: The Cylinder braiding of the quantum Weyl group of $sl_2$

abstract:
  Is is shown that the quantum Weyl group of $sl_2$ contains an element that is
a cylinder twist, i.e. it gives rise to representations of the braid group of
Coxeter type B.


title: Extensions of conformal modules

abstract:
  In this paper we classify extensions between irreducible finite conformal
modules over the Virasoro algebra, over the current algebras and over their
semidirect sums.


title: Integrals for braided Hopf algebras

abstract:
  Let H be a Hopf algebra in a rigid braided monoidal category with split
idempotents. We prove the existence of integrals on (in) H characterized by the
universal property, employing results about Hopf modules, and show that their
common target (source) object Int H is invertible. The fully braided version of
Radford's formula for the fourth power of the antipode is obtained. Connections
of integration with cross-product and transmutation are studied. The results
apply to topological Hopf algebras, e.g. a torus with a hole, which do not have
additive structure.


title: A Star Product for Complex Grasmann Manifolds

abstract:
  We explicitly construct a star product for the complex Grassmann manifolds
using the method of phase space reduction. Functions on $\mathbb{C}^{(p+q)\cdot
p~*}$, the space of $(p+q)\times p$ matrices of rank p, invariant under the
right action of $Gl(p,\mathbb{C})$ can be regarded as functions on the
Grassmann manifold $G_{p,q}(\mathbb{C})$, but do not form a subalgebra whereas
functions only invariant under the unitary subgroup $U(p)\subset
Gl(p,\mathbb{C})$ do. The idea is to construct a projection from $U(p)$- onto
$Gl(p,\mathbb{C})$-invariant functions, whose kernel is an ideal. This
projection can be used to define a star-algebra on $G_{p,q}(\mathbb{C})$ onto
which this projection acts as an algebra-epimorphism.


title:
$\sigma$-models on the quantum group manifolds $SL_{q}(2,R)$,
  $SL_{q}(2,R)/U_{h}(1)$, $C_{q}(2|0)$ and infinitesimal trasformations

abstract:
  The differential and variational calculus on the $SL_{q}(2,R)$ group is
constructed. The spontaneous breaking symmetry in the WZNW model with
$SL_{q}(2,R)$ quantum group symmetry and in the $\sigma$-models with
${SL_{q}(2,R)/U_{h}(1)}$ ,$C_{q}(2|0)$ quantum group symmetry is considered.
The Lagrangian formalism over the quantum group manifolds is discussed. The
classical solution of $C_{q}(2|0)$ {$\sigma$}-model is obtained.


title: Three Short Distance Structures from Quantum Algebras

abstract:
  We review known and we present new results on three types of short distance
structures of observables which typically appear in studies of quantum group
related algebras. In particular, one of the short distance structures is shown
to suggest a new mechanism for the introduction of internal symmetries.


title:
Dynamically twisted algebra $A_{q,p;\hat{\pi}}(\hat{gl_2})$ as current
  algebra generalizing screening currents of q-deformed Virasoro algebra

abstract:
  In this paper, we propose an elliptic algebra $A_{q,p;\hat{\pi}}(\hat{gl_2})$
which is based on the relations $RLL=LLR^{*}$, where $R$ and $R^{*}$ are the
dynamical R-maxtrices of $A^{(1)}_{1}$ type face model with the elliptic moduli
shifted by the center of the algebra. From the Ding-Frenkel correspondence, we
find that its corresponding (Drinfeld) current algebra at level one is the
algebra of screening currents for q-deformed Virasoro algebra.We realize the
elliptic algebra at level one by Miki's construction from the bosonization for
the type I and type II vertex operators.We also show that the algebra
$A_{q,p;\hat{\pi}}(\hat{gl_2})$ is related with the algebra
$A_{q,p}(\hat{gl_2})$ by a dynamically twisting.


title: Quantum and braided group Riemannian geometry

abstract:
  We formulate quantum group Riemannian geometry as a gauge theory of quantum
differential forms. We first develop (and slightly generalise) classical
Riemannian geometry in a self-dual manner as a principal bundle frame
resolution and a dual pair of canonical forms. The role of Levi-Civita
connection is naturally generalised to connections with vanishing torsion and
cotorsion, which we introduce. We then provide the corresponding quantum group
and braided group formulations with the universal quantum differential
calculus. We also give general constructions for examples, including quantum
spheres and quantum planes.


title:
The Quantized Knizhnik-Zamolodchikov Equation in Tensor Products of
  Irreducible sl(2)-Modules

abstract:
  We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ)
with values in a tensor product of irreducible sl(2) modules, the equation
defined in terms of rational R-matrices. We solve the equation in terms of
multidimensional q-hypergeometric integrals. We identify the space of solutions
of the qKZ equation with the tensor product of the corresponding modules over
the quantum group $U_qsl(2)$. We compute the monodromy of the qKZ equation in
terms of the trigonometric R-matrices.


title: Formal distribution algebras and conformal algebras

abstract:
  Conformal algebra is an axiomatic description of the operator product
expansion (or rather its Fourier transform) of chiral fields in a conformal
field theory. This is a review of recent developments in the subject.


title: 8-Vertex Correlation Functions and Twist Covariance of q-KZ Equation

abstract:
  We study the vertex operators $\Phi(z)$ associated with standard quantum
groups. The element $Z = RR^{t}$ is a "Casimir operator" for quantized
Kac-Moody algebras and the quantum Knizhnik-Zamolodchikov (q-KZ) equation is
interpreted as the statement $:Z\Phi(z): = \Phi(z)$. We study the covariance of
the q-KZ equation under twisting, first within the category of Hopf algebras,
and then in the wider context of quasi Hopf algebras. We obtain the
intertwining operators associated with the elliptic R-matrix and calculate the
two-point correlation function for the eight-vertex model.


title: Feynman Integral, Knot Invariant and Degree Theory of Maps

abstract:
  The universal Vassiliev invariant from the perturbative Chern-Simons theory
is actually a knot invariant without any correction term. The anomaly
considered by Bott and Taubes is proved to be zero.


title: Meromorphic tensor categories

abstract:
  We introduce the notion of meromorphic tensor category and illustrate it in
several examples. They include representations of quantum affine algebras,
chiral algebras of Beilinson and Drinfeld, G-vertex algebras of Borcherds, and
representations of GL over a local field. Hopefully the formalism will
accomodate various tensor structures arising in relation to the quantized
Knizhnik-Zamolodchikov equations and deformed CFT


title: Conjectured enumeration of Vassiliev invariants

abstract:
  A rational Ansatz is proposed for the generating function $\sum_{j,k}
\beta_{2j+k,2j}x^j y^k$, where $\beta_{m,u}$ is the number of primitive chinese
character diagrams with $u$ univalent and $2m-u$ trivalent vertices. For
$P_m:=\sum_{u\ge2}\beta_{m,u}$, the conjecture leads to the sequence
$$1,1,1,2,3,5,8,12,18,27,39,55,\underline{78,108,150,207,284,388,532,726}$$ for
primitive chord diagrams of degrees $m\le20$, with predictions underlined. The
asymptotic behaviour $\lim_{m\to\infty}P_m/r^m= 1.06260548918755$ results, with
$r=1.38027756909761$ solving $r^4=r^3+1$. Vassiliev invariants of knots are
then enumerated by $$0,1,1,3,4,9,14,27,44,
80,132,232,\underline{384,659,1095,1851,3065,5128,8461,14031}$$ and Vassiliev
invariants of framed knots by $$1,2,3,6,10,19,33,60,104,184,316,
548,\underline{932,1591,2686,4537,7602,12730,21191,35222}$$ These conjectures
are motivated by successful enumerations of irreducible Euler sums. Predictions
for $\beta_{15,10}$, $\beta_{16,12}$ and $\beta_{19,16}$ suggest that the
action of sl and osp Lie algebras, on baguette diagrams with ladder insertions,
fails to detect an invariant in each case.


title:
Quantum Orthogonal Planes: ISO_{q,r}(N) and SO_{q,r}(N) -- Bicovariant
  Calculi and Differential Geometry on Quantum Minkowski Space

abstract:
  We construct differential calculi on multiparametric quantum orthogonal
planes in any dimension N. These calculi are bicovariant under the action of
the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do
contain dilatations.
  If we require bicovariance only under the quantum orthogonal group
SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its
coordinates x^a, differentials dx^a and partial derivatives \partial_a without
the need of dilatations, thus generalizing known results to the multiparametric
case. Using real forms that lead to the signature (n+1,m) with m = n-1, n, n+1
, we find ISO_{q,r}(n+1, m) and SO_{q,r}(n+1,m) bicovariant calculi on the
multiparametric quantum spaces. The particular case of the quantum Minkowski
space ISO_{q,r}(3,1)/SO_{q,r}(3,1) is treated in detail.
  The conjugated partial derivatives \partial_a* can be expressed as linear
combinations of the \partial_a. This allows a deformation of the phase-space
where no additional operators (besides x^a and p_a) are needed.


title: What is a vertex algebra?

abstract:
  These are the notes of an informal talk in Bonn describing how to define an
analogue of vertex algebras in higher dimensions.


title:
Dual pairs and tensor categories of modules over Lie algebras
  gl_{\infty} and W_{1 +\infty}

abstract:
  We introduce a tensor category O_+ (resp. O_{-}) of certain modules of
gl_{\infty} with non-negative (resp. non-positive) integral central charges
with the usual tensor product. We also introduce a tensor category O_f
consisting of certain modules over GL(N) for all N. We show that the tensor
categories O_+, O_{-} and O_f are semisimple abelian and all equivalent to each
other. We give a formula to decompose a tensor product of two modules in each
of these categories. We also introduce a tensor category O^w of certain modules
over W_{1 +\infty} with non-negative integral central charges. We show that O^w
is semisimple abelian and give an explicit formula to decompose a tensor
product of two modules in O^w.


title: Representations of the q-deformed algebra $U'_q(so_{2,2})$

abstract:
  The main aim of this paper is to give classes of irreducible infinite
dimensional representations and of irreducible $*$-representations of the
q-deformed algebra $U'_q(so_{2,2})$ which is a real form of the non-standard
deformation $U'_q(so_4)$ of the universal enveloping algebra $U(so(4,C))$.
These representations are described by two complex parameters and are obtained
by "analytical continuation" of the irreducible finite dimensional
representations of the algebra $U'_q(so_4)$ in the basis corresponding to the
reduction from $U'_q(so_4)$ to $U(so_2\oplus so_2)$.


title: q-deformed algebras $U_q(so_n)$ and their representations

abstract:
  For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in
terms of trilinear relations for generating elements, most general finite
dimensional irreducible representations directly corresponding to those of
nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only
integers or only half-integers as in highest weights of the latter) are given
explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$
not equal to a root of unity, that representation operators indeed satisfy
relevant (trilinear) relations and define finite dimensional irreducible
representations is presented. The results show perfect suitability of the
Gel'fand-Tsetlin formalism concerning (nonstandard) $q$-deformation of $so(n)$.


title: A coset-type construction for the deformed Virasoro algebra

abstract:
  An analog of the minimal unitary series representations for the deformed
Virasoro algebra is constructed using vertex operators of the quantum affine
algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic
algebra $A_{q,p}(\hat{sl}_2)$.


title: Universal R-matrix Of The Super Yangian Double DY(gl(1|1))

abstract:
  Based on Drinfeld realization of super Yangian Double DY(gl(1|1)), its
pairing relations and universal R-matrix are given. By taking evaluation
representation of universal R-matrix, another realization $L^{\pm}(u)$ of
DY(gl(1|1)) is obtained. These two realizations of DY(gl(1|1)) are related by
the supersymmetric extension of Ding-Frenkel map.


title: On Fusion Algebras and Modular Matrices

abstract:
  We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal
field theories, affine Kac-Moody algebras at positive integer level, and
quantum groups at roots of unity. Using properties of the modular matrix $S$,
we find small sets of primary fields (equivalently, sets of highest weights)
which can be identified with the variables of a polynomial realization of the
$A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$
and level $k$, the number of these variables is the minimum possible, and we
conjecture that it is in fact minimal for most $r$ and $k$. We also find new,
systematic sources of zeros in the modular matrix $S$. In addition, we obtain a
formula relating the entries of $S$ at fixed points, to entries of $S$ at
smaller ranks and levels. Finally, we identify the number fields generated over
the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.


title: Deformation quantization of Poisson manifolds, I

abstract:
  I prove that every finite-dimensional Poisson manifold X admits a canonical
deformation quantization. Informally, it means that the set of equivalence
classes of associative algebras close to the algebra of functions on X is in
one-to-one correspondence with the set of equivalence classes of Poisson
structures on X modulo diffeomorphisms. In fact, a more general statement is
proven ("Formality conjecture"), relating the Lie superalgebra of polyvector
fields on X and the Hochschild complex of the algebra of functions on X.
Coefficients in explicit formulas for the deformed product can be interpreted
as correlators in a topological open string theory, although I do not use
explicitly the language of functional integrals. One of corollaries is a
justification of the orbit method in the representation theory.


title: Quantum Galois theory for finite groups

abstract:
  We prove the Galois correspondence between the subgroups of a finite
automorphism group G of a simple vertex operator algebra V and the vertex
operator subalgebras of V containing the set V^G of G-invariants.


title: Kazhdan-Lusztig polynomials and canonical basis

abstract:
  In this paper we show that the Kazhdan-Lusztig polynomials (and, more
generally, parabolic KL polynomials) for the group $S_n$ coincide with the
coefficients of the canonical basis in $n$th tensor power of the fundamental
representation of the quantum group $U_q sl_k$. We also use known results about
canonical bases for $U_q sl_2$ to get a new proof of recurrent formulas for KL
polynomials for maximal parabolic subgroups (geometrically, this case
corresponds to Grassmanians), due to Lascoux-Schutzenberger and Zelevinsky.


title: Hochschild cohomology and characteristic classes for star-products

abstract:
  We show that the Hochschild cohomology of the algebra obtained by formal
deformation quantization on a symplectic manifold is isomorphic to the formal
series with coefficients in the de Rham cohomology of the manifold. The
cohomology class obtained by differentiating the star-product with respect to
the deformation parameter is seen to be closely related to the characteristic
class of the quantization. A fundamental role in the analysis is played by
``quantum Liouville operators,'' which rescale the deformation parameter in the
same way in which Liouville vector fields scale the Poisson structure (or the
units of action). Several examples are given.


title: Harmonic Oscillator in Characteristic p

abstract:
  We construct an irreducible representation of the canonical commutation
relations by operators on a certain Banach space over a local field of
characteristic p. The Carlitz polynomials forming the basis of the space are
shown to be the counterparts of the Hermite functions for this situation. The
analogues of coherent states are related to the Carlitz exponential.


title: On shelling and flag vectors

abstract:
  This note defines a flag vector for $i$-graphs. The construction applies to
any finite combinatorial object that can be shelled. Two possible connections
to quantum topology are mentioned. Further details appear in the author's "On
quantum topology, hypergraphs and flag vectors", (preprint q-alg/9708001).


title: Topological Interpretations of Lattice Gauge Field Theory

abstract:
  We construct lattice gauge field theory based on a quantum group on a lattice
of dimension 1. Innovations include a coalgebra structure on the connections,
and an investigation of connections that are not distinguishable by
observables. We prove that when the quantum group is a deformation of a
connected algebraic group (over the complex numbers), then the algebra of
observables forms a deformation quantization of the ring of characters of the
fundamental group of the lattice with respect to the corresponding algebraic
group. Finally, we investigate lattice gauge field theory based on quantum
SL(2,C), and conclude that the algebra of observables is the Kauffman bracket
skein module of a cylinder over a surface associated to the lattice.


title: A master identity for homotopy Gerstenhaber algebras

abstract:
  We produce a master identity {m}{m}=0 for homotopy Gerstenhaber algebras, as
defined by Getzler and Jones and utilized by Kimura, Voronov, and Zuckerman in
the context of topological conformal field theories. To this end, we introduce
the notion of a "partitioned multilinear map" and explain the mechanics of
composing such maps. In addition, many new examples of pre-Lie algebras and
homotopy Gerstenhaber algebras are given.


title:
The Algebraic Structure of the $gl(n|m)$ Color Calogero-Sutherland
  Models

abstract:
  We extend the study on the algebraic structure of the $su(n)$ color
Calogero-Sutherland models to the case of $gl(n|m)$ color CS model and show
that the generators of the super-Yangian $Y(gl(n|m))$ can be obtained from two
$gl(n|m)$ loop algebras. Also, a super $W_{\infty}$ algebra for the SUSY CS
model is constructed.


title: Vector Fields and Differential Operators: Noncommutative Case

abstract:
  A notion of Cartan pairs as an analogy of vector fields in the realm of
noncommutative geometry has been proposed in q-alg/9609011 In this paper we
give an outline of the construction of a noncommutative analogy of the algebra
of partial differential operators as well as its natural (Fock type)
representation. We shall also define co-universal vector fields and covariant
derivatives.


title:
Branching functions of $A_{n-1}^{(1)}$ and Jantzen-Seitz problem for
  Ariki-Koike algebras

abstract:
  We study the restrictions of simple modules of Ariki-Koike algebras
$\H_m(\v)$ with set of parameters $\v= (\zeta;\zeta^{v_0},...
,\zeta^{v_{l-1}})$, where $\zeta$ is an $n$th root of unity, to their
subalgebras $\H_{m-j}(\v)$. Using a theorem of Ariki and the crystal basis
theory of Kashiwara, we relate this problem to the calculation of tensor
product multiplicities of highest weight irreducible representations of the
affine Lie algebra $A_{n-1}^{(1)}$. These multiplicities have a combinatorial
description in terms of higher level paths or highest-lift multipartitions.
This enables us to solve the Jantzen-Seitz problem for Ariki-Koike algebras,
that is, to determine which irreducible representations of $\H_m(\v)$ restrict
to irreducible representations of $\H_{m-1}(\v)$. From a combinatorial point of
view, this problem is identical to that of computing the tensor product of an
$A_{n-1}^{(1)}$-module of level $l$ and one of level 1. We also consider
natural generalisations of the Jantzen-Seitz problem corresponding to the
product of a level $l$ module by a level $l'>1$ module, and from the
commutativity of tensor products, we deduce a remarkable symmetry between the
generalised Jantzen-Seitz conditions and the sets of parameters of the
Ariki-Koike algebras.


title: Quantized Affine Algebras and Crystals with Head

abstract:
  Motivated by the work of Nakayashiki on the inhomogeneous vertex models of
6-vertex type, we introduce the notion of crystals with head. We show that the
tensor product of the highest weight crystal of level k and the perfect crystal
of level l is isomorphic to the tensor product of the perfect crystal of level
l-k and the highest weight crystal of level k.


title: Quantum Orthogonal Cayley-Klein Groups and Algebras

abstract:
  The extension of FRT quantization theory for the nonsemisimple CK groups is
suggested. The quantum orthogonal CK groups are realized as the Hopf algebras
of the noncommutative functions over an associative algebras with nilpotent
commutative generators. The quantum CK algebras are obtained as the dual
objects to the corresponding quantum groups.


title: Braided Deformations of Monoidal Categories and Vassiliev Invariants

abstract:
  Braided deformations of (symmetric) monoidal categories are related to
Vassiliev theory by a direct generalization of well-known results relating
"quantum" knot invariants to Vassiliev invariants. The deformation theory of
braidings is subsumed by the deformation theory of monoidal functors, which
proves surprisingly rich: the deformation complex of a monoidal functor has the
same structure as the deformation complex of an algebra, including a pre-Lie
structure, from which it is see that the problem of deforming monoidal functors
(including braidings) is purely cohomological in nature.


title: Effective generalized Seifert-Van Kampen: how to calculate $\Omega X$

abstract:
  Suppose $X$ is a 1-connected simplicial set with finitely many nondegenerate
simplices. We give an effective algorithm to calculate a simplicial set with
the $n$-type of the loop space $\Omega X$. Iterating gives an algorithm to
calculate the $\pi_i(X)$, different from the algorithms already known due to E.
Brown and Kan-Curtis. The method is an effective version of the generalized
Seifert-Van Kampen theorem of alg-geom/9704006. This can be viewed as a Van
Kampen statement concerning the loop space $\Omega X$ with its delooping
structure. We use Segal's delooping machinery but at the end we speculate on
extensions to other delooping machinery.


title: Lie Algebras in Fock Space

abstract:
  A catalogue of explicit realizations of representations of (super) Lie
algebras and quantum algebras in Fock space is presented.

categories: q-alg cond-mat hep-th math-ph math.MP math.QA


title: The zero divisors of the Cayley-Dickson algebras over the real numbers

abstract:
  In this paper we describe algebraically the zero divirsors of the Cayley-
Dickson algebras $\a_{n}=\erre^{2^n}$ for $n \ge 4$ over the real numbers.


title: An Introduction to the Noncommutative Sphere and some Extensions

abstract:
  This is a copy of the talk given at the conference ``Methods in Field
Theory'' at Stara Lesna, The Slovak Republic, Sepemeber 22-26, 1997. An
introduction to the noncommutative sphere and a summary of the results of
articles q-alg/9703038 \cite{Gratus5}, and q-alg/9708003 \cite{Gratus6} is
given. This includes results about the the algebra of scalar, spinor and vector
fields on the noncommutative sphere. Possible extensions of these results
including a ``Wick rotation'' to the one and two sheeted hyperboloid are also
examined.


title: Orthogonal Polynomials of Types A and B and Related Calogero Models

abstract:
  Calogero-Sutherland models associated to the Weyl groups of type A and B with
exchange terms included in the Hamiltonians systems have non-symmetric
eigenfunctions, which are products of the ground state with members of a family
of orthogonal polynomials. These polynomials can be defined and studied by
using the differential-difference operators introduced by the author in TAMS
1989 (311), 167-183. There is a study of polynomials which are invariant or
alternating for parabolic subgroups of the symmetric group. The detailed
analysis depends on using two bases of polynomials, one of which transforms
monomially under group actions and the other one is orthogonal. There are
formulas for norms and point-evaluations which are simplifications of those of
Sahi. For any parabolic subgroup of the symmetric group there is a skew
operator on polynomials which leads to evaluation at (1,1,...,1) of the
quotient of the unique skew polynomial in a given irreducible subspace by the
minimum alternating polynomial, analogously to a Weyl character formula. The
last section concerns orthogonal polynomials for the type B Weyl group with an
emphasis on the Hermite-type polynomials. A complete basis of eigenfunctions of
Yamamoto's B_N spin Calogero model is obtained by multiplying these polynomials
by the ground state.


title:
Unitary Representations and BRST Structure of the Quantum Anti--de
  Sitter Group at Roots of Unity

abstract:
  It is shown that for suitable roots of unity, there exist finite--dimensional
unitary representations of $U_q(so(2,3))$ corresponding to all classical
one-particle representations with (half)integer spin, with the correct
low-energy limit. In the massless case for spin $\geq 1$, a subspace of "pure
gauges'' appears which must be factored out, as classically. Unitary
many-particle representations are defined, with the same low-energy states as
classically. Furthermore, a remarkable element of the center of $U_q(so(2,3))$
is identified which plays the role of the BRST operator, for any spin. The
corresponding ghosts are an intrinsic part of indecomposable representations.


title:
Rank one lattice type vertex operator algebras and their automorphism
  groups

abstract:
  Let L be a positive definite even lattice of rank one and V_L^+ be the fixed
points of the lattice VOA V_L associated to L under an automorphism of V_L
lifting the -1$ isometry of L. A set of generators and the full automorphism
group of V_L^+ are determined.


title: spl(p,q) superalgebra and differential operators

abstract:
  Series of finite dimensional representations of the superalgebras spl(p,q)
can be formulated in terms of linear differential operators acting on a
suitable space of polynomials. We sketch the general ingredients necessary to
construct these representations and present examples related to spl(2,1) and
spl(2,2). By revisiting the products of projectivised representations of sl(2),
we are able to construct new sets of differential operators preserving some
space of polynomials in two or more variables. In particular, this allows to
express the representation of spl(2,1) in terms of matrix differential
operators in two variables. The corresponding operators provide the building
blocks for the construction of quasi exactly solvable systems of two and four
equations in two variables. We also present a quommutator deformation of
spl(2,1) which, by construction, provides an appropriate basis for analyzing
the quasi exactly solvable systems of finite difference equations.


title:
Dirac operator on $\kappa$-Minkowski space bicovariant differential
  calculus and deformed U(1) gauge theory

abstract:
  Derivation of $\kappa$-Poincare bicovariant commutation relations between
coordinates and 1-forms on $\kappa$-Minkowski space is given using Dirac
operator and Allain Connes formula. The deformed U(1) gauge theory and
appearance of an additional spin 0 gauge field is discussed.


title:
Hecke algebras as subalgebras of Clifford geometric algebras of
  multivectors

abstract:
  Clifford geometric algebras of multivectors are introduced which exhibit a
bilinear form which is not necessarily symmetric. Looking at a subset of
bi-vectors in CL(K^{2n},B), we proof that theses elements generate the Hecke
algebra H_{K}(n+1,q) if the bilinear form B is chosen appropriately. This
shows, that q-quantization can be generated by Clifford multivector objects
which describe usually composite entities. This contrasts current approaches
which give deformed versions of Clifford algebras by deforming the one-vector
variables. Our example shows, that it is not evident from a mathematical point
of view, that q-deformation is in any sense more elementary than the undeformed
structure.


title: $d^N=0$

abstract:
  We study the generalized homology associated with a nilpotent endomorphism
$d$ satisfying $d^N=0$. For simplicial modules we construct such nilpotent
endomorphisms and we prove a general result relating the corresponding
generalized homologies to the ordinary homology. We also discuss the
generalization of the notion of graded differential algebra in this context.


title: Harmonic Analysis on the quantum Lorentz group

abstract:
  This work begins with a review of complexification and realification of Hopf
algebras. We emphasize the notion of multiplier Hopf algebras for the
description of different classes of functions (compact supported, bounded,
unbounded) on complex quantum groups and the construction of the associated
left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$
with complex spins, we give a new description of the unitary representations of
$SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q
(2,\CC)_{\RR}$. The major theorem of this article is the Plancherel theorem for
the Quantum Lorentz Group.


title: Level one representations of $U_q(G_2^{(1)})$

abstract:
  We construct level one representations of the quantum affine algebra
$U_q(G_2^{(1)})$ by vertex operators from bosonic fields.


title: q-Deforming Maps for Lie Group Covariant Heisenberg Algebras

abstract:
  We briefly summarize our systematic construction procedure of q-deforming
maps for Lie group covariant Weyl or Clifford algebras.


title: A Burge tree of Virasoro-type polynomial identities

abstract:
  Using a summation formula due to Burge, and a combinatorial identity between
partition pairs, we obtain an infinite tree of q-polynomial identities for the
Virasoro characters \chi^{p, p'}_{r, s}, dependent on two finite size
parameters M and N, in the cases where:
  (i) p and p' are coprime integers that satisfy 0 < p < p'.
  (ii) If the pair (p', p) has a continued fraction (c_1, c_2, ... , c_{t-1},
c_t+2), where t >= 1, then the pair (s, r) has a continued fraction (c_1, c_2,
... , c_{u-1}, d), where 1 =< u =< t, and 1 =< d =< c_{u}.
  The limit M -> infinity, for fixed N, and the limit N -> infinity, for fixed
M, lead to two independent boson-fermion-type q-polynomial identities: in one
case, the bosonic side has a conventional dependence on the parameters that
characterise the corresponding character. In the other, that dependence is not
conventional. In each case, the fermionic side can also be cast in either of
two different forms. Taking the remaining finite size parameter to infinity in
either of the above identities, so that M -> infinity and N -> infinity, leads
to the same q-series identity for the corresponding character.


title: The deformed Virasoro algebra at roots of unity

abstract:
  We discuss some aspects of the representation theory of the deformed Virasoro
algebra $\virpq$. In particular, we give a proof of the formula for the Kac
determinant and then determine the center of $\virpq$ for $q$ a primitive N-th
root of unity. We derive explicit expressions for the generators of the center
in the limit $t=qp^{-1}\to \infty$ and elucidate the connection to the
Hall-Littlewood symmetric functions. Furthermore, we argue that for
$q=\sqrtN{1}$ the algebra describes `Gentile statistics' of order $N-1$, i.e.,
a situation in which at most $N-1$ particles can occupy the same state.


title: Deformations of the classical W-algebras associated to D_n, E_6 and G_2

abstract:
  We give explicit formulas for the generators of $q$-deformed W-algebras
associated to Lie algebras $D_n, E_6$ and $G_2$, and compute the Poisson
brackets between the generators.


title:
Lie Bialgebra Structures for Centrally Extended Two- Dimensional Galilei
  Algebra and their Lie-Poisson Counterparts

abstract:
  All bialgebra structures for centrally extended Galilei algebra are
classified. The corresponding Lie-Poisson structures on centrally extended
Galilei group are found.


title: Positivity of Dunkl's intertwining operator

abstract:
  For a finite reflection group on $\b R^N,$ the associated Dunkl operators are
parametrized first-order differential-difference operators which generalize the
usual partial derivatives. They generate a commutative algebra which is - under
weak assumptions - intertwined with the algebra of partial differential
operators by a unique linear and homogeneous isomorphism on polynomials. In
this paper it is shown that for non-negative parameter values, this
intertwining operator is positivity-preserving on polynomials and allows a
positive integral representation on certain algebras of analytic functions.
This result in particular implies that the generalized exponential kernel of
the Dunkl transform is positive-definite.


title: Cartan-Weyl basis for Quantum Affine Superalgebra U_q(^osp(1|2))

abstract:
  Cartan-Weyl basis for the quantum affine superalgebra U_q(^osp(1|2)) is
constructed in an explicit form.


title: Degenerate Double Affine Hecke Algebra And Conformal Field Theory

abstract:
  We introduce a class of induced representations of the degenerate double
affine Hecke algebra of gl_N and analyze their structure mainly by means of
intertwiners. We also construct them from modules of the affine Lie algebra
using Knizhnik-Zamolodchikov connections in the conformal field theory. This
construction provides a natural quotient of induced modules, which turns out to
be the unique irreducible one under a certain condition. Some cunjectual
formulas are presented for the symmetric part of these quotients.


title: Nonsymmetric Koornwinder polynomials and duality

abstract:
  Motivated by the work of Koornwinder, Macdonald, Cherednik, Noumi, and van
Diejen we define a 6-parameter double affine Hecke algebra and establish its
basic structural properties, including the existence of an involution. We
relate the algebra to (symmetric and nonsymmetric) Koornwinder polynomials via
the method of intertwiners and, as a consequence, obtain a proof of Macdonald's
duality conjecture for Koornwinder polynomials. Combined with earlier work of
van Diejen this completely settles all the outstanding conjectures of Macdonald
and Koornwinder about these polynomials.


title:
Boundary solutions of the quantum Yang-Baxter equation and solutions in
  three dimensions

abstract:
  Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to
be those in the boundary of (but not in) the variety of solutions to the
``modified'' qYB equation, the latter being analogous to the modified classical
Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to
the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary
solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last
section we list and give quantizations for all classical r-matrices in $sl(3)
\wedge sl(3)$.


title: Differential Calculus on Fuzzy Sphere and Scalar Field

abstract:
  We find that there is an alternative possibility to define the chirality
operator on the fuzzy sphere, due to the ambiguity of the operator ordering.
Adopting this new chirality operator and the corresponding Dirac operator, we
define Connes' spectral triple on the fuzzy sphere and the differential
calculus. The differential calculus based on this new spectral triple is
simplified considerably. Using this formulation the action of the scalar field
is derived.


title: Duality in infinite dimensional Fock representations

abstract:
  We construct and study in detail various dual pairs acting on some Fock
representations between a finite dimensional Lie group and a completed infinite
rank affine algebra associated to an infinite affine Cartan matrix. We give
explicit decompositions of a Fock representation into a direct sum of
irreducible isotypic subspaces with respect to the action of a dual pair,
present explicit formulas for the common highest weight vectors and calculate
the corresponding highest weights. We further outline applications of these
dual pairs to the study of tensor products of modules of such an infinite
dimensional Lie algebra.


title:
Isomorphisms of type $A$ affine Hecke algebras and multivariable
  orthogonal polynomials

abstract:
  We examine two isomorphisms between affine Hecke algebras of type $A$
associated with parameters $q^{-1}$, $t^{-1}$ and $q$, $t$. One of them maps
the non-symmetric Macdonald polynomials $E_{\eta}(x;q^{-1},t^{-1})$ onto
$E_{\eta}(x;q,t)$, while the other maps them onto non-symmetric analogues of
the multivariable Al-Salam & Carlitz polynomials. Using the properties of
$E_{\eta}(x;q^{-1},t^{-1})$, the corresponding properties of these latter
polynomials can then be elucidated.


title: Lie algebras and degenerate Affine Hecke Algebras of type A

abstract:
  We construct a family of exact functors from the BGG category of
representations of the Lie algebra sl to the category of finite-dimensional
representations of the degenerate (or graded) affine Hecke algebra H of GL.
These functors transform Verma modules to standard modules or zero, and simple
modules to simple modules or zero. Any simple H-module can be thus obtained.


title:
3-State Potts model and automorphism of vertex operator algebra of order
  3

abstract:
  We define an automorphism of VOA of order 3 by using a sub VOA isomorphic to
a direct sum of 3-state Potts models $L(\ff,0)$ and an its module $L(\ff,3)$.
This automorphism is a 3A element of the monster simple group if $V$ is the
moonshine VOA $V^{\natural}$.


title: Quantization of the space of conformal blocks

abstract:
  We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated
to $gl(N)$, defined in terms of rational R-matrices. We prove that under
certain resonance conditions, the qKZ connection has a non-trivial invariant
subbundle which we call the subbundle of quantized conformal blocks. The
subbundle is given explicitly by algebraic equations in terms of the Yangian
$Y(gl(N))$ action. The subbundle is a deformation of the subbundle of conformal
blocks in CFT. The proof is based on an identity in the algebra with two
generators $x,y$ and defining relation $xy=yx+yy$.


title:
On algebraic equations satisfied by hypergeometric solutions of the qKZ
  equation

abstract:
  We consider the $sl(2)$ quantized Knizhnik-Zamolodchikov equation (qKZ),
defined in terms of rational R-matrices. The properties of the equation change
when the step of the equation takes a resonance value. In this case the
discrete connection defined by the qKZ equation has a invariant subbundle which
we call the subbundle of quantized conformal blocks. Solutions of the qKZ
equation were constructed in [TV1], [MV1] in terms of multidimensional
hypergeometric integrals. In this paper we show that for a resonance step all
hypergeometric solutions take values in the subbundle of quantized conformal
blocks, moreover the values span the subbundle of quantized conformal blocks
under certain conditions. We describe the space of hypergeometric solutions in
terms of the quantum group $U_qsl(2)$.


title: Universal R-matrix for null-plane quantized Poincar{\'e} algebra

abstract:
  The universal ${\cal R}$--matrix for a quantized Poincar{\'e} algebra ${\cal
P}(3+1)$ introduced by Ballesteros et al is evaluated. The solution is obtained
as a specific case of a formulated multidimensional generalization to the
non-standard (Jordanian) quantization of $sl(2)$.


title:
Vector Coherent State Realization of Representations of the Affine Lie
  Algebra $\hat{sl}(2)$

abstract:
  The method of vector coherent states is generalized to study representations
of the affine Lie algebra $\hat{sl}(2)$. A large class of highest weight irreps
is explicitly constructed, which contains the integrable highest weight irreps
as special cases.


title:
Discretizations of the Schr\"odinger equation with quantum algebra
  symmetry

abstract:
  Two quantum Hopf structures for the Schr\"odinger algebra as well as their
corresponding differential-difference realizations are presented. For each case
a (space or time) discretization of the Schr\"odinger equation is deduced and
the quantum Schr\"odinger generators are shown to be symmetry operators.


title:
Biorthogonal polynomials associated with reflection groups and a formula
  of Macdonald

abstract:
  Dunkl operators are differential-difference operators on $\b R^N$ which
generalize partial derivatives. They lead to generalizations of Laplace
operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on.
In this paper we introduce two systems of biorthogonal polynomials with respect
to Dunkl's Gaussian distributions in a quite canonical way. These systems,
called Appell systems, admit many properties known from classical Hermite
polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian
distributions. In particular, these polynomials lead to a new proof of a
generalized formula of Macdonald due to Dunkl.
  The ideas for this paper are taken from recent works on non-Gaussian white
noise analysis and from the umbral calculus.


title: Quantum SL(3,C)'s with classical representation theory

abstract:
  We study and classify almost all quantum SL(3,C)'s whose representation
theory is ``similar'' to that of the (ordinary) group SL(3,C). Only one case,
related to smooth elliptic curves, could not be treated completely.


title: $\tau$-function within group theory approach and its quantization

abstract:
  This is a review of the results related to generalizations of the notion of
$\tau$-function and integrable hierarchies and to their interpretation within
the group theory framework that admits an immediate quantization procedure.
Different group theory structures underlying integrable system are discussed in
detail as well as their quantum deformations.


title: A Factorization of the Conway Polynomial

abstract:
  A string link S can be closed in a canonical way to produce an ordinary
closed link L. We also consider a twisted closing which produces a knot K. We
give a formula for the Conway polynomial of L as a product of the Conway
polynomial of K times a power series whose coefficients are given as explicit
functions of the Milnor invariants of S. One consequence is a formula for the
first non-vanishing coefficient of the Conway polynomial of L in terms of the
Milnor invariants of L. There is an analogous factorization of the
multivariable Alexander polynomial.