title: Perfect Crystals and q-deformed Fock Spaces

abstract:
  A general scheme for the wedge construction of q-deformed Fock spaces using
the theory of perfect crystals is presented.
  Let $U_q(\g)$ be a quantum affine algebra. Let $V$ be a finite-dimensional
$U'_q(\g)$-module with a perfect crystal base of level~$l$. Let $V_\aff\simeq
V\otimes\C[z,z^{-1}]$ be the affinization of $V$, with crystal base
$(L_\aff,B_\aff)$. The wedge space $V_\aff\wedge V_\aff$ is defined as the
quotient of $V_\aff\otimes V_\aff$ by the subspace generated by the action of
$U_q(\g)[z^a\otimes z^b +z^b\otimes z^a]_{a,b\in\Z}$ on $v\otimes v$ ($v$ an
extremal vector). The wedge space $\bigwedge^r V_\aff$ ($r\in\N$) is defined
similarly. Normally ordered wedges are defined by using the energy function
$H:B_\aff\otimes B_\aff\to\Z$. Under certain assumptions, it is proved that
normally ordered wedges form a base of $\bigwedge^r V_\aff$.
  A q-deformed Fock space is defined as the inductive limit of $\bigwedge^r
V_\aff$ as $r\to\infty$, taken along the semi-infinite wedge associated to a
ground state sequence. It is proved that normally ordered wedges form a base of
the Fock space and that the Fock space has the structure of an integrable
$U_q(\g)$-module. An action of the bosons, which commute with the
$U'_q(\g)$-action, is given on the Fock space. It induces the decomposition of
the q-deformed Fock space into the tensor product of an irreducible
$U_q(\g)$-module and a bosonic Fock space.
  As examples, Fock spaces for types $A^{(2)}_{2n}$, $B^{(1)}_n$,
$A^{(2)}_{2n-1}$, $D^{(1)}_n$ and $D^{(2)}_{n+1}$ at level~1 and $A^{(1)}_1$ at
level~$k$ are constructed. The commutation relations of the bosons in each of
these cases are calculated, using two point functions of vertex operators.


title: Vectors and covectors in non-commutative setting

abstract:
  Following the guidelines of classical differential geometry the `building
material' for the tensor calculus in non-commutative geometry is suggested. The
algebraic account of moduli of vectors and covectors is carried out.


title: Integrality of two variable Kostka functions

abstract:
  The two variable Kostka functions are the scalar products of the Macdonald
polynomials with the Schur polynomials with respect to the scalar product which
makes the Hall-Littlewood polynomials pairwise orthogonal. A conjecture of
Macdonald states that they are polynomials in q and t with non-negative
integral coefficients. We prove that the Kostka functions are at least
polynomials with integral coefficients. The main idea is to prove an analogous
statement for the non-symmetric Macdonald polynomials by establishing recursion
relations via the affine Hecke algebra.


title: Symmetric and non-symmetric quantum Capelli polynomials

abstract:
  We define a family of symmetric and a family of non-symmetric polynomials in
terms of vanishing conditions. These families depend on two paramters, q and t.
Their main feature is that they consist of non-homogeneous polynomials. The
symmetric polynomials form the quantized version of polynomials occuring in the
context of generalized Capelli identities. We show that these quantum Capelli
polynomials are also characterized by q-difference equations. More precisely,
they are eigenfunctions of Cherednik type operators and transform under an
affine Hecke algebra. Thus, we are able to identify their top homogeneous
component as a Macdonald polynomial (symmetric or non-symmetric, respectively).


title:
Hom functor and the associativity of tensor products of modules for
  vertex operator algebras

abstract:
  We give a new, construction-free proof of the associativity of tensor product
for modules for rational vertex operator algebras under certain convergence
conditions.


title: A supergeometric interpretation of vertex operator superalgebras

abstract:
  Huang's geometric interpretation of vertex operator algebras is extended to a
supergeometric interpretation of vertex operator superalgebras. In particular,
the geometry of spheres with punctures and local analytic coordinates in terms
of exponentials of derivations is extended to the geometry of superspheres with
punctures, a given spin structure, and local superconformal coordinates in
terms of exponentials of superderivations. The notion of supergeometric vertex
operator superalgebra is introduced, and in addition, the notion of
(superalgebraic) vertex operator superalgebra over a Grassmann algebra and with
odd formal variables is introduced. The main result is that the category of
supergeometric vertex operator superalgebras over a Grassmann algebra with a
given spin structure and the category of vertex operator superalgebras over a
Grassmann algebra are isomorphic. In addition, the supergeometric symmetry
between the two different possible spin structures naturally gives rise to a
symmetry in the superalgebraic structure.


title: Some Remarks on Quantum and Braided Group Gauge Theory

abstract:
  We clarify some aspects of quantum group gauge theory and its recent
generalisations (by T. Brzezinski and the author) to braided group gauge theory
and coalgebra gauge theory. We outline the diagrammatic version of the braided
case. We study the bosonisation of any braided group provides as a trivial
principal bundle in three ways.


title:
A central extension of $\cD Y_{\hbar}(\gtgl_2)$ and its vertex
  representations

abstract:
  A central extension of $\cD Y_{\hbar}(\gtgl_2)$ is proposed. The bosonization
of level $1$ module and vertex operators are also given.


title:
Bosonic representations of Yangian Double $\cD Y_{\hbar}(\gtg)$ with
  $\gtg=\gtgl_N,\gtsl_N$

abstract:
  On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the
Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension.
The Gauss decomposition of T-matrices gives us the so-called Drinfel'd
generators. Using these generators, we present some examples of both finite and
infinite dimensional representations that are quite natural deformations of the
corresponding affine counterpart.


title: Localization of modules over small quantum groups

abstract:
  A report on the works hep-th/9411050, q-alg/9412017, q-alg/9503013,
q-alg/9506011 and a joint work with R.Bezrukavnikov.


title:
A system of difference equations with elliptic coefficients and Bethe
  vectors

abstract:
  An elliptic analogue of the $q$ deformed Knizhnik-Zamolodchikov equations is
introduced. A solution is given in the form of a Jackson-type integral of Bethe
vectors of the XYZ-type spin chains.


title:
Commuting difference operators with applications to orthogonal
  polynomials in several variables

abstract:
  Elementary properties of the Koornwinder-Macdonald multivariable Askey-Wilson
polynomials are discussed. Studied are the orthogonality, the difference
equations, the recurrence relations, and the orthonormalization constants for
these polynomials. Essential in our approach are certain commuting difference
operators simultaneously diagonalized by the polynomials.


title:
Properties of some families of hypergeometric orthogonal polynomials in
  several variables

abstract:
  Limiting cases are studied of the Koornwinder-Macdonald multivariable
generalization of the Askey-Wilson polynomials. We recover recently and not so
recently introduced families of hypergeometric orthogonal polynomials in
several variables consisting of multivariable Wilson, continuous Hahn and
Jacobi type polynomials, respectively. For each class of polynomials we provide
systems of difference (or differential) equations, recurrence relations, and
expressions for the norms of the polynomials in terms of the norm of the
constant polynomial.


title:
The Universal R-Matrix, Burau Representaion and the Melvin-Morton
  Expansion of the Colored Jones Polynomial

abstract:
  P. Melvin and H. Morton studied the expansion of the colored Jones polynomial
of a knot in powers of q-1 and color. They conjectured an upper bound on the
power of color versus the power of q-1. They also conjectured that the bounding
line in their expansion generated the inverse Alexander-Conway polynomial.
These conjectures were proved by D. Bar-Natan and S. Garoufalidis.
  We have conjectured that other `lines' in the Melvin-Morton expansion are
generated by rational functions with integer coefficients whose denominators
are powers of the Alexander-Conway polynomial. Here we prove this conjecture by
using the R-matrix formula for the colored Jones polynomial and presenting the
universal R-matrix as a `perturbed' Burau matrix.


title: SL(2,C) Topological Quantum Field Theory with Corners

abstract:
  The paper contains the construction of a topological quantum field theory in
which gluings along surfaces with boundary are allowed. The construction is
made by using quantum deformations of sl(2,C). In the construction there
appears a sign obstruction to some of the Moore-Seiberg equations. This problem
is solved by means of the Klein four group.


title: Quantization of Poisson groups -- II

abstract:
  Let $ G^\tau $ be a connected simply connected semisimple algebraic group,
endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $
H^\tau $ be its dual Poisson group. By means of Drinfeld's double construction
and dualization via formal Hopf algebras, we construct new quantum groups $
U_{q,\phi}^M ({\frak h}) $ --- dual of $ U_{q,\phi}^{M'} ({\frak g}) $ ---
which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau $; we study
their specializations at roots of 1 (in particular, their classical limits),
thus discovering new quantum Frobenius morphisms. The whole description dualize
for $ H^\tau $ what was known for $ G^\tau $, completing the quantization of
the pair $ (G^\tau,H^\tau) $.


title: Universal $R$-matrix for non-standard quantum $sl(2,\R)$

abstract:
  A universal $R$-matrix for the non-standard (Jordanian) quantum deformation
of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter
equation is obtained from some finite dimensional representations of this Lie
bialgebra quantization of $sl(2,\R)$.


title: Geometrical Meaning of R-matrix Action for Quantum Groups at Roots of 1

abstract:
  The present work splits in two parts: first, we perform a straightforward
generalization of results from [Re], proving autoquasitriangularity of quantum
groups $ U_q(\frak{g}) $ and their unrestricted specializations at roots of 1,
in particular the function algebra $ F[H] $ of the Poisson group $ H $ dual of
$ G $; second, as a main contribution, we prove the convergence of the
(specialized) $ R $--matrix action to a birational automorphism of a
2$\ell$--fold ramified covering of the specialization of $ U_q(\frak{g}) $ at a
primitive $ \ell $--th root of 1, and of a 2-fold ramified covering of $ H $,
thus giving a geometric content to the notion of triangularity (or
autoquasitriangularity) for quantum groups.


title:
Primitive Vassiliev Invariants and Factorization in Chern-Simons
  Perturbation Theory

abstract:
  The general structure of the perturbative expansion of the vacuum expectation
value of a Wilson line operator in Chern-Simons gauge field theory is analyzed.
The expansion is organized according to the independent group structures that
appear at each order. It is shown that the analysis is greatly simplified if
the group factors are chosen in a certain way that we call canonical. This
enables us to show that the logarithm of a polinomial knot invariant can be
written in terms of primitive Vassiliev invariants only.


title:
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and
  Quantum Affine Algebras

abstract:
  The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra $sl_2$ is a system of linear difference
equations with values in a tensor product of $sl_2$ Verma modules. We solve the
equation in terms of multidimensional $q$-hypergeometric functions and define a
natural isomorphism between the space of solutions and the tensor product of
the corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter
$q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$.
  We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric $R$-matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy
group of the differential Knizhnik-Zamolodchikov equation.
  In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.


title: Bethe Ansatz for Higher Spin XYZ Models --- Low-lying Excitations ---

abstract:
  A higher spin generalization of the XYZ spin chain defined by the fusion
procedure is considered. Energy and momentum of the ground state and low-lying
excited states of the model are computed by means of an isomorphism between a
space of theta functions on which the Sklyanin algebra acts and a space
obtained by the fusion procedure.


title: Understanding the Kauffman bracket skein module

abstract:
  The Kauffman bracket skein module $K(M)$ of a 3-manifold $M$ is defined over
formal power series in the variable $h$ by letting $A=e^{h/4}$. For a compact
oriented surface $F$, it is shown that $K(F \times I)$ is a quantization of the
$\g$-characters of the fundamental group of $F$, corresponding to a
geometrically defined Poisson bracket. Finite type invariants for unoriented
knots and links are defined. Topologically free Kauffman bracket modules are
shown to generate finite type invariants. It is shown for compact $M$ that
$K(M)$ can be generated as a module by cables on a finite set of knots.
Moreover, if $M$ contains no incompressible surfaces, the module is finitely
generated.


title:
Rings of $SL_2({\mathbb C})$-Characters and the Kauffman Bracket Skein
  Module

abstract:
  Let $M$ be a compact orientable 3-manifold. The set of characters of
$SL_2({\mathbb C})$ representations of the fundamental group of $M$ forms a
closed affine algebraic set. We show that its coordinate ring is isomorphic to
a specialization of the Kauffman bracket skein module modulo its nilradical.
This is accomplished by making the module into a combinatorial analog of the
ring, in which tools of skein theory are exploited to illuminate relations
among characters. We conclude with an application, proving that a small
manifold's specialized module is necessarily finite dimensional.


title: A Higher-level Bailey Lemma

abstract:
  We propose a generalization of Bailey's lemma, useful for proving $q$-series
identities. As an application, generalizations of Euler's identity, the
Rogers-Ramanujan identities, and the Andrews-Gordon identities are derived.
This generalized Bailey lemma also allows one to derive identities for the
branching functions of higher-level $A^{(1)}_1$ cosets.


title:
Jackson Integral Representations of Modified $q$-Bessel Functions and
  $q$-Bessel-Macdonald Functions

abstract:
  The $q$ analog of Modified Bessel functions and Bessel-Macdonald functions,
were defined in our previous work (q-alg/950913) as general solutions of a
second order difference equations. Here we present a collection of their
representations by the Jackson q-integral.


title: Representation Theory of Lattice Current Algebras

abstract:
  Lattice current algebras were introduced as a regularization of the left- and
right moving degrees of freedom in the WZNW model. They provide examples of
lattice theories with a local quantum symmetry $U_q(\sg)$. Their representation
theory is studied in detail. In particular, we construct all irreducible
representations along with a lattice analogue of the fusion product for
representations of the lattice current algebra. It is shown that for an
arbitrary number of lattice sites, the representation categories of the lattice
current algebras agree with their continuum counterparts.


title: Knot energies and knot invariants

abstract:
  This an article about some elementary geometric and combinatorial natures of
various knot energies. A related "new" knot invariant -- the X-crossing number
-- is introduced.


title: Exact two-spinon dynamical correlation function of the Heisenberg model

abstract:
  We derive the exact contribution of two spinons to the dynamical correlation
function of the spin-1/2 Heisenberg model. For this, we use the isotropic
limits of the exact form factors that have been recently computed through the
quantum affine symmetry of the anisotropic Heisenberg model $XXZ$


title: Wakimoto construction in the principal gradation

abstract:
  It is well known that the bosonized version of the Wakimoto construction
allows the explicit realization of any affine algebra $\widehat{g}$, with
arbitrary level $k$ in the homogeneous gradation, in terms of $dim(g)$ free
bosonic fields. In this paper, we show in the case of the simplest affine
algebra $\widehat{sl(2)}$, that the bosonized Wakimoto realization can be
extended to the principal gradation only when $k$ is equal to the critical
level, i.e., -2. In this case, this construction can be achieved in terms of
arbitrary number (larger than 1) of free bosonic fields.


title:
On the relation between $U_q(\widehat{sl(2)})$ vertex operators and
  $q$-zonal functions

abstract:
  We show how the states constructed from the action of the modes of bosonized
vertex operators, that intertwine $U_q(\widehat{sl(2)})$ modules, are related
to $q$-zonal functions.


title: The algebraic structure of relative twisted vertex operators

abstract:
  Twisted vertex operators based on rational lattices have had many
applications in vertex operator algebra theory and conformal field theory. In
this paper, ``relativized'' twisted vertex operators are constructed in a
general context based on isometries of rational lattices, and a generalized
twisted Jacobi identity is established for them. This result generalizes many
previous results. Relatived untwisted vertex operators had been studied in a
monograph by the authors. The present paper includes as a special case the
proof of the main relations among twisted vertex operators based on even
lattices announced some time ago by the second author.


title:
Vertex Operators of the $q$-Virasoro Algebra; Defining Relations,
  Adjoint Actions and Four Point Functions

abstract:
  Primary fields of the $q$-deformed Virasoro algebra are constructed.
Commutation relations among the primary fields are studied. Adjoint actions of
the deformed Virasoro current on the primary fields are represented by the
shift operator $\Theta_{\xi} f(x)=f(\xi x)$. Four point functions of the
primary fields enjoy the connection formula associated with the Boltzmann
weights of the fusion Andrews-Baxter-Forrester model.


title: Differential calculus on `non-standard' ($h$-deformed) Minkowski spaces

abstract:
  The differential calculus on `non-standard' $h$-Minkowski spaces is given. In
particular it is shown that, for them, it is possible to introduce coordinates
and derivatives which are simultaneously hermitian.


title: Quasi-Exactly Soluble Potentials and Deformed Oscillators

abstract:
  It is proved that quasi-exactly soluble potentials corresponding to an
oscillator with harmonic, quartic and sextic terms, for which the $n+1$ lowest
levels of a given parity can be determined exactly, may be approximated by WKB
equivalent potentials corresponding to deformed anharmonic oscillators of
SU$_q$(1,1) symmetry, which have been used for the description of vibrational
spectra of diatomic molecules. This connection allows for the immediate
approximate determination of the levels of the same parity lying above the
lowest $n+1$ known levels, as well as of all levels of the opposite parity.
Such connections are not possible in the cases of the q-deformed oscillator,
the Q-deformed oscillator, and the modified P\"oschl-Teller potential with
SU(1,1) symmetry.


title: The multiple gamma functions and the multiple q-gamma functions

abstract:
  We give an asymptotic expansion (the higher Stirling formula) and an infinite
product representation (the Weierstrass product representation) of the
Vign\'{e}ras multiple gamma functions by considering the classical limit of the
multiple q-gamma functions.


title:
Universal weight systems and the Melvin-Morton expansion of the colored
  Jones knot invariant

abstract:
  We study the asymptotic expansion of the colored Jones polynomial (the
Melvin-Morton expansion) using a recursion formula for the deframed universal
weight system for the $sl(2)$ Lie algebra. Combined with the formula for the
universal weight system for the Lie superalgebra $gl(1|1)$ (which corresponds
to the Alexander-Conway knot polynomial) this formula gives a very short proof
of the Melvin-Morton conjecture relating the colored Jones invariant and the
Alexander-Conway polynomial of knots.


title: Affine Hecke algebras and raising operators for Macdonald polynomials

abstract:
  We introduce certain raising and lowering operators for Macdonald polynomials
(of type $A_{n-1}$) by means of Dunkl operators. The raising operators we
discuss are a natural $q$-analogue of raising operators for Jack polynomials
introduced by L.Lapointe and L.Vinet. As an application we prove the
integrality of double Kostka coefficients. Double analog of the multinomial
coefficients are introduced.


title:
$q$-Difference raising operators for Macdonald polynomials and the
  integrality of transition coefficients

abstract:
  We study certain $q$-difference raising operators for Macdonald polynomials
(of type $A_{n-1}$) which are originated from the $q$-difference-reflection
operators introduced in our previous paper. These operators can be regarded as
a $q$-difference version of the raising operators for Jack polynomials
introduced by L.Lapointe and L.Vinet. As an application of our $q$-difference
raising operators we give an elementary proof of the integrality of the double
Kostka coefficients which had been conjectured I.G. Macdonald. We also
determine their quasi-classical limits, which give rise to (differental)
raising operators for Jack polynomials.


title: First-Order Differential Calculi Over Multi-Braided Quantum Groups

abstract:
  A differential calculus of the first order over multi-braided quantum groups
is developed. In analogy with the standard theory, left/right-covariant and
bicovariant differential structures are introduced and investigated.
Furthermore, antipodally covariant calculi are studied. The concept of the
*-structure on a multi-braided quantum group is formulated, and in particular
the structure of left-covariant *-covariant calculi is analyzed. A special
attention is given to differential calculi covariant with respect to the action
of the associated braid system. In particular it is shown that the left/right
braided-covariance appears as a consequence of the left/right-covariance
relative to the group action. Braided counterparts of all basic results of the
standard theory are found.


title: General Frame Structures On Quantum Principal Bundles

abstract:
  A noncommutative-geometric generalization of the classical formalism of frame
bundles is developed, incorporating into the theory of quantum principal
bundles the concept of the Levi-Civita connection. The construction of a
natural differential calculus on quantum principal frame bundles is presented,
including the construction of the associated differential calculus on the
structure group. General torsion operators are defined and analyzed.
Illustrative examples are presented.


title: Quantum Principal Bundles and Their Characteristic Classes

abstract:
  A brief exposition of the general theory of characteristic classes of quantum
principal bundles is given. The theory of quantum characteristic classes
incorporates ideas of classical Weil theory into the conceptual framework of
non-commutative differential geometry. A purely cohomological interpretation of
the Weil homomorphism is given, together with a standard geometrical
interpretation via quantum invariant polynomials. A natural spectral sequence
is described. Some quantum phenomena appearing in the formalism are discussed.


title: Quantum Classifying Spaces and Universal Quantum Characteristic Classes

abstract:
  A construction of the noncommutative-geometric counterparts of classical
classifying spaces is presented, for general compact matrix quantum structure
groups. A quantum analogue of the classical concept of the classifying map is
introduced and analyzed. Interrelations with the abstract algebraic theory of
quantum characteristic classes are discussed. Various non-equivalent approaches
to defining universal characteristic classes are outlined.


title:
Quantum Gauge Transformations and Braided Structure on Quantum Principal
  Bundles

abstract:
  It is shown that every quantum principal bundle is braided, in the sense that
there exists an intrinsic braid operator twisting the functions on the bundle.
A detailed algebraic analysis of this operator is performed. In particular, it
turns out that the braiding admits a natural extension to the level of
arbitrary differential forms on the bundle. Applications of the formalism to
the study of quantum gauge transformations are presented.


title: General methods for constructing bispectral operators

abstract:
  We present methods for obtaining new solutions to the bispectral problem. We
achieve this by giving its abstract algebraic version suitable for
generalizations. All methods are illustrated by new classes of bispectral
operators.


title: A Fedosov Star Product of Wick Type for K\"ahler Manifolds

abstract:
  In this letter we compute some elementary properties of the Fedosov star
product of Weyl type, such as symmetry and order of differentiation. Moreover,
we define the notion of a star product of Wick type on every K\"ahler manifold
by a straight forward generalization of the corresponding star product in
$\mathbb C^n$: the corresponding sequence of bidifferential operators
differentiates its first argument in holomorphic directions and its second
argument in antiholomorphic directions. By a Fedosov type procedure we give an
existence proof of such star products for any K\"ahler manifold.


title:
(Shifted) Macdonald Polynomials: q-Integral Representation and
  Combinatorial Formula

abstract:
  We extend some results about shifted Schur functions to the general context
of shifted Macdonald polynomials. We obtain two explicit formulas for these
polynomials: a $q$-integral representation and a combinatorial formula. Our
main tool is a $q$-integral representation for ordinary Macdonald polynomials.
We also discuss duality for shifted Macdonald polynomials and Jack degeneration
of these polynomials.


title: Intertwining Operators of Double Affine Hecke Algebras

abstract:
  A systematic study of the representation theory of double affine Hecke
algebras and related harmonic analysis is started in this paper. Continuing the
previous papers we use the technique of intertwining operators to create
Macdonald polynomials, estimate their denominators, generalize the classical
representations of p-adic affine Hecke algebras in the spaces of functions on
affine Weyl groups, and to find out when induced representations are
irreducible and co-spherical. The connection with recent results by Sahi and
Knop on the integrality of the Macdonald polynomials is established.


title: Centre and Representations of U_q(sl(2|1)) at Roots of Unity

abstract:
  Quantum groups at roots of unity have the property that their centre is
enlarged. Polynomial equations relate the standard deformed Casimir operators
and the new central elements. These relations are important from a physical
point of view since they correspond to relations among quantum expectation
values of observables that have to be satisfied on all physical states. In this
paper, we establish these relations in the case of the quantum Lie superalgebra
U_q(sl(2|1)). In the course of the argument, we find and use a set of
representations such that any relation satisfied on all the representations of
the set is true in U_q(sl(2|1)). This set is a subset of the set of all the
finite dimensional irreducible representations of U_q(sl(2|1)), that we
classify and describe explicitly.


title: Group-like elements in quantum groups, and Feigin's conjecture

abstract:
  Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf
n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody
algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$
subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion
(with respect to the natural grading) of the quantized enveloping algebra of
${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let
$P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject
to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k<l\le m$) where
$C=(C_{ij})=(d_ia_{ij})$ is the symmetric matrix corresponding to $A$. We
construct a group-like element ${\bf e}_{\bf i}\in P_{\bf i}\bigotimes \hat
U_q({\bf n})$. This element gives rise to the evaluation homomorphism
$\psi_{\bf i}:{\bf C}_q[N]\to P_{\bf i}$ given by $\psi_{\bf i}(x)=x({\bf
e}_{\bf i})$, where ${\bf C}_q[N]=U_q({\bf n})^0$ is the restricted dual of
$U_q({\bf n})$. Under a well-known isomorphism of algebras ${\bf C}_q[N]$ and
$U_q({\bf n})$, the map $\psi_{\bf i}$ identifies with Feigin's homomorphism
$\Phi({\bf i}): U_q({\bf n})\to P_{\bf i}$. We prove that the image of
$\psi_{\bf i}$ generates the skew-field of fractions ${\cal F}(P_{\bf i})$ if
and only if ${\bf i}$ is a reduced expression of some element $w$ in the Weyl
group $W$; furthermore, in the latter case, ${\rm Ker}~\psi_{\bf i}$ depends
only on $w$ (so we denote $I_w:={\rm Ker}~\psi_{\bf i}$). This result
generalizes the results in [5], [6] to the case of Kac-Moody algebras. We also
construct an element ${\cal R}_w\in \big({\bf C}_q[N]/I_w\big)\bigotimes \hat
U_q({\bf n})$ which specializes to ${\bf e}_{\bf i}$ under the embedding ${\bf
C}_q[N]/I_w\hookrightarrow P_{\bf i}$. The elements ${\cal R}_w$ are closely


title:
A family of quantum projective spaces and related q-hypergeometric
  orthogonal polynomials

abstract:
  We define a one-parameter family of two-sided coideals in U_q(gl(n)) and
study the corresponding algebras of infinitesimally right invariant functions
on the quantum unitary group U_q(n). The Plancherel decomposition of these
algebras with respect to the natural transitive U_q(n)-action is shown to be
the same as in the case of a complex projective space. By computing the radial
part of a suitable Casimir operator, we identify the zonal spherical functions
(i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional
irreducible representations) as Askey-Wilson polynomials containing two
continuous and one discrete parameter. In certain limit cases, the zonal
spherical functions are expressed as big and little q-Jacobi polynomials
depending on one discrete parameter.


title: On the uniqueness of the Moyal structure of phase-space functions

abstract:
  It is shown that the only associative algebras with a trivial center defined
on functions of $\Rl^N$ by an integral kernel are generalized Moyal algebras,
corresponding to some particular operator ordering. Similarly, the only such
Lie algebras are generalized Moyal or Poisson Lie algebras. In both cases these
structures are isomorphic respectively to the Moyal algebra, the Moyal Lie
algebra and the Poisson Lie algebra. These results are independent of $N$,
which is necessarily even, and generalize previous work on the subject.


title: Vassiliev Invariants and Gleam Polynomials

abstract:
  It was shown by Goussarov that Vassiliev invariants are polynomials in the
gleams for a fixed Turaev shadow. In this paper we show that Vassiliev
invariants are almost characterized by this fact. We also prove that the space
of knot invariants which are polynomials in the gleams is bigger than the
Vassiliev subspace.


title: Scasimir operator, Scentre and Representations of U_q(osp(1|2))

abstract:
  A bosonic operator of U_q(osp(1|2)) that anticommutes with the fermionic
generators appears to be useful to describe the relations in the centre of
U_q(osp(1|2)) for q a root of unity (in the unrestricted specialisation). As in
the classical case, it also simplifies the classification of finite dimensional
irreducible representations.


title: On Casimir's Ghost

abstract:
  We define on the universal enveloping superalgebra of osp(1|2n) a nonstandard
adjoint action, endowing it with a module structure. This allows, in
particular, to construct a bosonic operator which anticommutes with all the
fermionic generators and which appears to be the square root of a certain
Casimir operator.


title: Deformed boson algebras and the quantum double construction

abstract:
  The quantum double construction of a $q$-deformed boson algebra possessing a
Hopf algebra structure is carried out explicitly. The $R$-matrix thus obtained
is compared with the existing literature.


title: Burau representation and random walk on string links

abstract:
  Using a probabilistic interpretation of the Burau representation of the braid
group offered by Vaughan Jones, we generalize the Burau representation to a
representation of the semigroup of string links. This representation is
determined by a linear system, and is dominated by finite type string link
invariants. For positive string links, the representation matrix can be
interpreted as the transition matrix of a Markov process. For positive
non-separable links, we show that all states are persistent. (This is an
extensively revised version of the first author's original paper titled "Burau
representation and random walk on knots".)


title:
Algebraic Bethe ansatz for the elliptic quantum group
  $E_{\tau,\eta}(sl_2)$

abstract:
  To each representation of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is
associated a family of commuting transfer matrices. We give common eigenvectors
by a version of the algebraic Bethe ansatz method. Special cases of this
construction give eigenvectors for IRF models, for the eight-vertex model and
for the two-body Ruijsenaars operator. The latter is a $q$-deformation of
Hermite's solution of the Lam\'e equation.


title: Quantum Lie algebras; their existence, uniqueness and $q$-antisymmetry

abstract:
  Quantum Lie algebras are generalizations of Lie algebras which have the
quantum parameter h built into their structure. They have been defined
concretely as certain submodules of the quantized enveloping algebras. On them
the quantum Lie bracket is given by the quantum adjoint action.
  Here we define for any finite-dimensional simple complex Lie algebra g an
abstract quantum Lie algebra g_h independent of any concrete realization. Its
h-dependent structure constants are given in terms of inverse quantum
Clebsch-Gordan coefficients. We then show that all concrete quantum Lie
algebras are isomorphic to an abstract quantum Lie algebra g_h.
  In this way we prove two important properties of quantum Lie algebras: 1) all
quantum Lie algebras associated to the same g are isomorphic, 2) the quantum
Lie bracket of any quantum Lie algebra is $q$-antisymmetric. We also describe a
construction of quantum Lie algebras which establishes their existence.


title: Introduction to Quantum Lie Algebras

abstract:
  Quantum Lie algebras are generalizations of Lie algebras whose structure
constants are power series in $h$. They are derived from the quantized
enveloping algebras $\uqg$. The quantum Lie bracket satisfies a generalization
of antisymmetry. Representations of quantum Lie algebras are defined in terms
of a generalized commutator.
  In this paper the recent general results about quantum Lie algebras are
introduced with the help of the explicit example of $(sl_2)_h$.


title:
A new class of deformed special functions from quantum homogeneous
  spaces

abstract:
  We study the most elementary aspects of harmonic analysis on a homogeneous
space of a deformation of the two-dimensional Euclidean group, admitting
generalizations to dimensions three and four, whose quantum parameter has the
physical dimensions of a length. The homogeneous space is recognized as a new
quantum plane and the action of the Euclidean quantum group is used to
determine an eigenvalue problem for the Casimir operator, that constitutes the
analogue of the Schroedinger equation in the presence of such deformation. The
solutions are given in the plane wave and in the angular momentum bases and are
expressed in terms of hypergeometric series with non commuting parameters


title: Melvin-Morton conjecture and primitive Feynman diagrams

abstract:
  We give a very short proof of the Melvin-Morton conjecture relating the
colored Jones polynomial and the Alexander polynomial of knots. The proof is
based on the explicit evaluation of the corresponding weight systems on
primitive elements of the Hopf algebra of chord diagrams which, in turn,
follows from simple identities between four-valent tensors on the Lie algebra
$sl_2$ and the Lie superalgebra $gl(1|1)$. This shows that the miraculous
connection between the Jones and Alexander invariants follows from the
similarity (supersymmetry) between $sl_2$ and $gl(1|1)$.


title:
Bosonization of Admissible Representations of $U_q(\widehat{sl}_2)$ at
  Level $-\frac{1}{2}$ and $q$-Vertex Operators

abstract:
  We construct a representation of $U_q(\widehat{sl}_2)$ at level $-1/2$ by
using the bosonic Fock spaces. The irreducible modules are obtained as the
kernel of a certain operator, in contrast to the construction by Feingold and
Frenkel for $q=1$ where such a procedure is not necessary. We also bosonize the
$q$-vertex operators associated with the vector representation.


title: Separation of variables for Gaudin-Calogero systems

abstract:
  We construct an elliptic analogue of Sklyanin's separation of variables for
the $sl(2)$ Gaudin system, using an adaptation of Drinfeld's Radon
transformations.


title:
(2+1) null-plane quantum Poincar\'e group from a factorized universal
  $R$-matrix

abstract:
  The non-standard (Jordanian) quantum deformations of $so(2,2)$ and (2+1)
Poincar\'e algebras are constructed by starting from a quantum $sl(2,\R)$ basis
such that simple factorized expressions for their corresponding universal
$R$-matrices are obtained. As an application, the null-plane quantum (2+1)
Poincar\'e Poisson-Lie group is quantized by following the FRT prescription.
Matrix and differential representations of this null-plane deformation are
presented, and the influence of the choice of the basis in the resultant
$q$-Schr\"odinger equation governing the deformed null plane evolution is
commented.


title:
Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra
  $A(V_{L})$

abstract:
  It is proved that Zhu's algebra for vertex operator algebra associated to a
positive-definite even lattice of rank one is a finite-dimensional
semiprimitive quotient algebra of certain associative algebra introduced by
Smith. Zhu's algebra for vertex operator algebra associated to any
positive-definite even lattice is also calculated and is related to a
generalization of Smith's algebra.


title: q-Ultraspherical polynomials for q a root of unity

abstract:
  Properties of the $q$-ultraspherical polynomials for $q$ being a primitive
root of unity are derived using a formalism of the $so_q(3)$ algebra. The
orthogonality condition for these polynomials provides a new class of
trigonometric identities representing discrete finite-dimensional analogs of
$q$-beta integrals of Ramanujan.


title:
Zeros and orthogonality of the Askey-Wilson polynomials for q a root of
  unity

abstract:
  We study some properties of the Askey-Wilson polynomials (AWP) when q is a
primitive N-th root of unity. For general four-parameter AWP, zeros of the N-th
polynomial and the orthogonality measure are found explicitly. Special
subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi
polynomials, are considered in detail. A set of discrete weight functions
positive on a real interval is described. Some new trigonometric identities
related to the AWP are obtained. Normalization conditions of some polynomials
are expressed in terms of the Gauss sums.


title: Squared Hopf algebras and reconstruction theorems

abstract:
  Given an abelian k-linear rigid monoidal category V, where k is a perfect
field, we define squared coalgebras as objects of cocompleted V tensor V
(Deligne's tensor product of categories) equipped with the appropriate notion
of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are
defined without use of braiding.
  If V is the category of k-vector spaces, squared (co)algebras coincide with
conventional ones. If V is braided, a braided Hopf algebra can be obtained from
a squared one.
  Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras
in V and corresponding fibre functors to V (which is not the case with other
definitions). Finally, squared quasitriangular Hopf coalgebra is a solution to
the problem of defining quantum groups in braided categories.


title: Casson-Lin's invariant of a knot and Floer homology

abstract:
  A. Casson defined an intersection number invariant which can be roughly
thought of as the number of conjugacy classes of irreducible representations of
$\pi_1(Y)$ into $SU(2)$ counted with signs, where $Y$ is an oriented integral
homology 3-sphere. X.S. Lin defined an similar invariant (signature of a knot)
to a braid representative of a knot in $S^3$. In this paper, we give a natural
generalization of the Casson-Lin's invariant to be (instead of using the
instanton Floer homology) the symplectic Floer homology for the representation
space (one singular point) of $\pi_1(S^3 \setminus K)$ into $SU(2)$ with
trace-free along all meridians. The symplectic Floer homology of braids is a
new invariant of knots and its Euler number of such a symplectic Floer homology
is the negative of the Casson-Lin's invariant.


title: On Quantum Groups Co-Representations

abstract:
  We carry out a generalization of quantum group co-representations in order to
encode in this structure those cases where non-commutativity between
endomorphism matrix entries and quantum space coordinates happens.


title: On the Deformation Quantization of super-Poisson Brackets

abstract:
  We show that for every vector bundle E over any given symplectic manifold M
there exists an explicitly given super-Poisson bracket on the space of sections
of the dual Grassmann bundle associated to E built out of symplectic structure
of M, a fibre metric on E, and a connection in E compatible with the given
fibre metric. Moreover, we construct a deformation quantization for this space
of sections by means of a Fedosov type procedure.


title: Traces of Intertwining Operators for the Yangian Double

abstract:
  The traces over infinite dimensional representations of the central extended
Yangian double for the product of operators which intertwine these
representations are calculated. For the special combinations of the
intertwining operators the traces are identified with form factors of local
operators in SU(2)-invariant Thirring model. This identification is based on
the identities which are deformed analogs of the Gauss-Manin connection
identities for the hyperelliptic curves.


title:
q-Krawtchouk polynomials as spherical functions on the Hecke algebra of
  type B

abstract:
  The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group
of type B, contains the generic Hecke algebra for the symmetric group, i.e. the
Weyl group of type A, as a subalgebra. Inducing the index representation of the
subalgebra gives a Hecke algebra module, which splits multiplicity free. The
corresponding zonal spherical functions are calculated in terms of q-Krawtchouk
polynomials. The result covers a number of previously established
interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group,
finite groups of Lie type, hypergroups and the quantum SU(2) group. Jimbo's
analogue of the Frobenius-Schur-Weyl duality is a key ingredient in the proof.


title: Representation theory of deformed oscillator algebras

abstract:
  The representation theory of deformed oscillator algebras, defined in terms
of an arbitrary function of the number operator~$N$, is developed in terms of
the eigenvalues of a Casimir operator~$C$. It is shown that according to the
nature of the $N$ spectrum, their unitary irreducible representations may fall
into one out of four classes, some of which contain bosonic, fermionic or
parafermionic Fock-space representations as special cases. The general theory
is illustrated by classifying the unitary irreducible representations of the
Arik-Coon, Chaturvedi-Srinivasan, and Tamm-Dancoff oscillator algebras, which
may be derived from the boson one by the recursive minimal-deformation
procedure of Katriel and Quesne. The effects on non-Fock-space representations
of the minimal deformation and of the quommutator-commutator transformation,
considered in such a procedure, are studied in detail.


title: Shifted Schur Functions

abstract:
  The classical algebra $\Lambda$ of symmetric functions has a remarkable
deformation $\Lambda^*$, which we call the algebra of shifted symmetric
functions. In the latter algebra, there is a distinguished basis formed by
shifted Schur functions $s^*_\mu$, where $\mu$ ranges over the set of all
partitions. The main significance of the shifted Schur functions is that they
determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal
enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$.
  The functions $s^*_\mu$ are closely related to the factorial Schur functions
introduced by Biedenharn and Louck and further studied by Macdonald and other
authors.
  A part of our results about the functions $s^*_\mu$ has natural classical
analogues (combinatorial presentation, generating series, Jacobi--Trudi
identity, Pieri formula). Other results are of different nature (connection
with the binomial formula for characters of $GL(n)$, an explicit expression for
the dimension of skew shapes $\lambda/\mu$, Capelli--type identities, a
characterization of the functions $s^*_\mu$ by their vanishing properties,
`coherence property', special symmetrization map $S(\frak{gl}(n))\to
U(\frak{gl}(n))$.
  The main application that we have in mind is the asymptotic character theory
for the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\to\infty$.
  The results of this paper were used in \cite{Ok1--3}.


title: A new construction of the semi-infinite BGG resolution

abstract:
  We introduce the techniques of semiregular bimodules over a Lie algebra with
respect to a Lie subalgebra. Using this techniques in the case of affine Lie
algebras we introduce twisting functors on the categories of modules. These
functors are enumerated by elements of the affine Weyl group corresponding to
the chosen affine Lie algebra and map Verma modules to so called twisted Verma
modules. Applying twisting functors to the BGG resolution of an integrable
simple module we obtain a sequence of complexes called the twisted BGG
resolutions. We show that the twisted BGG resolutions form an inductive system
of complexes. It turns out that the limit complex of the inductive system is
exactly the semi-infinite BGG resolution introduced by B. Feigin and E. Frenkel
and consisting of direct sums of Wakimoto modules.


title: Quantum double of a (locally) compact group

abstract:
  We generalise the quantum double construction of Drinfel'd to the case of the
(Hopf) algebra of suitable functions on a compact or locally compact group. We
will concentrate on the *-algebra structure of the quantum double. If the
conjugacy classes in the group are countably separated, then we classify the
irreducible *-representations by using the connection with so-called
transformation group algebras. For finite groups, we will compare our
description to the result of Dijkgraaf, Pasquier and Roche. Finally we will
work out the explicit examples of SU(2) and SL(2,R).


title:
Free Field Representation of Level-$k$ Yangian Double ${\cal
  D}Y(sl_2)_k$ and Deformation of Wakimoto Modules

abstract:
  Free field representation of level-$k (\not=0,-2)$ Yangian double ${\cal
D}Y(sl_2)_k$ and a corresponding deformation of Wakimoto modules are presented.
We also realize two types of vertex operators intertwining these modules.


title: Some automorphisms of Generalized Kac-Moody algebras

abstract:
  To any symmetry of the Cartan matrix of a Generalized Kac-Moody (GKM) algebra
we associate a family of automorphisms of the algebra which act in a natural
way on the modules of the GKM algebra. We introduce the twining character of a
module which is obtained by inserting the action of such an automorphism into
the trace that appears in the ordinary character. Twining characters of
integrable highest weight modules and Verma modules satisfy a character formula
which involves a certain subgroup of the Weyl group. This subgroup is shown to
be the Weyl group of another GKM algebra, called the orbit Lie algebra, and
hence the twining characters coincide with ordinary characters of the orbit Lie
algebra.


abstract:
  A level-one representation of the quantum affine superalgebra
$\U_q(\hat{\frak{sl}}(M+1|N+1))$ and vertex operators associated with the
fundamental representations are constructed in terms of free bosonic fields.
Character formulas of level-one irreducible highest weight modules of
$\U_q(\hat{\frak{sl}}(2|1))$ are conjectured.


title:
Hecke symmetries and characteristic relations on Reflection Equation
  algebras

abstract:
  We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix)
influence the algebraic structure of the corresponding Reflection Equation (RE)
algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the
matrix of generators of the RE algebra related to a finite rank even Hecke
symmetry are derived.


title:
Irreducible tensor operators in the regular coaction formalisms of
  compact quantum group algebras

abstract:
  The defining conditions for the irreducible tensor operators associated with
the unitary irreducible corepresentions of compact quantum group algebras are
deduced first in both the right and left regular coaction formalisms. In each
case it is shown that there are {\em{two}} types of irreducible tensor
operator, which may be called `ordinary' and `twisted'. The consistency of the
definitions is demonstrated, and various consequences are deduced, including
generalizations of the Wigner-Eckart theorem for both the ordinary and twisted
operators. Also included are discussions (within the regular coaction
formalisms for compact quantum group algebras) of inner-products, basis
functions, projection operators, Clebsch-Gordan coefficients, and two types of
tensor product of corepresentations. The formulation of quantum homogeneous
spaces for compact quantum group algebras is discussed, and the defining
conditions for the irreducible tensor operators associated with such quan!
  tum homogeneous spaces and with the unitary irreducible corepresentions of
the compact quantum group algebras are then deduced. There are two versions,
which correspond to restrictions of the right and left regular coactions. In
each case it is again shown that there are ordinary and twisted irreducible
tensor operators. Various consequences are deduced, including the corresponding
generalizations of the Wigner-Eckart theorem.


title:
Abstract carrier space formalism for the irreducible tensor operators of
  compact quantum group algebras

abstract:
  Defining conditions for irreducible tensor operators associated with the
unitary irreducible corepresentations of compact quantum group algebras are
deduced within the framework of the abstract carrier space formalism. It is
shown that there are {\em{two}} types of irreducible tensor operator, which may
be called `ordinary' and `twisted'. The consistency of the definitions is
demonstrated, and various consequences are deduced, including generalizations
of the Wigner-Eckart theorem for both the ordinary and twisted operators.
Examples of irreducible tensor operators for the standard deformation of the
function algebra of the compact Lie group $SU(2)$ are described to demonstrate
the applicability of the new definitions.


title: Nonstandard Poincare and Heisenberg Algebras

abstract:
  New deformations of the Poincare group $Fun(P(1+1))$ and its dual enveloping
algebra $U(p(1+1))$ are obtained as a contraction of the $h$-deformed
(Jordanian) quantum group $Fun(SL_h(2))$ and its dual. A nonstandard
quantization of the Heisenberg algebra $U(h(1))$ is also investigated.


title: Equivalence of Two Approaches to the mKdV Hierarchies

abstract:
  The equivalence between the approaches of Drinfeld-Sokolov and Feigin-Frenkel
to the mKdV hierarchies is established. A new derivation of the mKdV equations
in the zero curvature form is given. Connection with the Baker-Akhiezer
function and the tau-function is also discussed.


title: Solutions of the elliptic qKZB equations and Bethe ansatz I

abstract:
  We give an integral representation for solutions of the elliptic quantum
Knizhnik-Zamolodchikov-Bernard difference equations, in the case of sl(2). The
result is based on a geometric construction of highest weight representations
of the elliptic quantum group associated to sl(2). We also obtain Bethe ansatz
eigenfunctions for the corresponding integrable systems of difference
operators, and prove their completeness in some cases.


title:
Quantization of geometry associated to the quantized
  Knizhnik-Zamolodchikov equations

abstract:
  It is known that solutions of the Knizhnik-Zamolodchikov differential
equations are given by integrals of closed differential forms over suitable
cycles. In this paper a quantization of this geometric construction is
described leading to solution of the quantized Knizhnik-Zamolodchikov
difference equations.


title: Semiquantum Geometry

abstract:
  In this paper we study associative algebras with a Poisson algebra structure
on the center acting by derivations on the rest of the algebra. These
structures, which we call Poisson fibred algebras, appear in the study of
quantum groups at roots of 1 and related algebras, as well as in the
representation theory of affine Lie algebras at the critical level. Poisson
fibred algebras lead to a generalization of Poisson geometry, which we develop
in the paper. We also take up the general study of noncommutative spaces which
are close to enough commutative ones so that they contain enough points to have
interesting commutative geometry. One of the most striking uses of our
noncommutative spaces is the quantum Borel-Weil-Bott Theorem for quantum sl_q
(2) at a root of unity, which comes as a calculation of the cohomology of
actual sheaves on actual topological spaces.


title: Factorial supersymmetric Schur functions and super Capelli identities

abstract:
  A factorial analogue of the supersymmetric Schur functions is introduced. It
is shown that factorial versions of the Jacobi--Trudi and Sergeev--Pragacz
formulae hold. The results are applied to construct a linear basis in the
center of the universal enveloping algebra for the Lie superalgebra gl(m|n) and
to obtain super-analogues of the higher Capelli identities.


title: Geometric Construction of Crystal Bases

abstract:
  We realize the crystal associated to the quantized enveloping algebras with a
symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the
cotangent bundle of the quiver variety. As a by-product, we give a
counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of
the characteristic variety of the intersection cohomology sheaves associated
with the Schubert cells of type A and also to the similar problem asked by
Lusztig on the characteristic variety of the perverse sheaves corresponding to
canonical bases.


title:
From quantum affine groups to the exact dynamical correlation function
  of the Heisenberg model

abstract:
  The exact form factors of the Heisenberg models $XXX$ and $XXZ$ have been
recently computed through the quantum affine symmetry of $XXZ$ model in the
thermodynamic limit. We use them to derive an exact formula for the
contribution of two spinons to the dynamical correlation function of $XXX$
model at zero temperature.


title:
Noncommutative geometry and its relation to stochastic calculus and
  symplectic mechanics

abstract:
  In the context of a noncommutative differential calculus on the algebra of
real valued functions of an $n$-dimensional manifold $M$, a commutative and
associative product of 1-forms is naturally defined. Ordinary differential
calculus corresponds to this product being trivially zero. We consider the
minimal generalization in which the algebra of 1-forms with this product is
nilpotent of degree 3. Basic tensor analysis and differential geometry are
developped in this context and applied to the formulation of symplectic
geometry and Hamiltonian dynamics. It is shown that the corresponding Liouville
equation can take the form of a generalized Fokker-Planck equation, well known
in statistical mechanics of open systems, as well as in stochastic calculus.
Specifically it may be the same with that obtained via Ito stochastic calculus,
when solving a linear stochastic differential equation with a Wiener process as
the stochastic term. The close connection between noncommutative differential
structures of this kind, with semimartingale stochastic processes on manifolds
thus suggested, is further explored.


title:
Link invariants from $N$-state vertex models: an alternative
  construction independent of statistical models

abstract:
  We reproduce the hierarchy of link invariants associated to the series of
$N$-state vertex models with a method different from the original construction
due to Akutsu, Deguchi and Wadati. The alternative method substitutes the
`crossing symmetry' property exhibited by the Boltzmann weights of the vertex
models by a similar property which, for the purpose of constructing link
invariants, encodes the same information but requires only the limit of the
Boltzmann weights when the spectral parameter is sent to infinity.


title: On the existence of deformed Lie-Poisson structures for quantized groups

abstract:
  The geometrical description of deformation quantization based on quantum
duality principle makes it possible to introduce deformed Lie-Poisson
structure. It serves as a natural analogue of classical Lie bialgebra for the
case when the initial object is a quantized group. The explicit realization of
the deformed Lie-Poisson structure is a difficult problem. We study the special
class of such constructions characterized by quite a simple form of tanjent
vector fields. It is proved that in such a case it is sufficient to find four
Lie compositions that form two deformations of the first order and four Lie
bialgebras. This garantees the existence of two families of deformed
Lie-Poisson structures due to the intrinsic symmetry of the initial
compositions. The explicit example is presented.


title:
Irreducible Representations of Jordanian Quantum Algebra ${\cal
  U}_h(sl(2))$ Via a Nonlinear Map

abstract:
  The generators of the Jordanian quantum algebra ${\cal U}_h(sl(2))$ are
expressed as nonlinear invertible functions of the classical $sl(2)$
generators. This permits immediate explicit construction of the finite
dimensional irreducible representations of the algebra ${\cal U}_h(sl(2))$.
Using this construction, new finite dimensional solutions of the Yang-Baxter
equation may be obtained.


title: A cellular braid action and the Yang - Baxter equation

abstract:
  Using a theorem of Schechtman - Varchenko on integral expressions for
solutions of Knizhnik - Zamolodchikov equations we prove that the solutions of
the Yang - Baxter equation associated to complex simple Lie algebras belong to
the class of generalised Magnus representations of the braid group. Hence they
can be obtained from the homology of a certain cell complex, or equivalently as
group homology of iterated free groups.


title: Lie Algebras and the Four Color Theorem

abstract:
  We present a ``reasonable'' statement about Lie algebras that is equivalent
to the Four Color Theorem. The notions appearing in the statement also appear
in the theory of finite-type invariants of knots (Vassiliev invariants) and
3-manifolds.


title: L'algebre de Hopf bitensorielle

abstract:
  Nous construisons pour tout corps k de caracteristique zero un foncteur de la
categorie des k-espaces vectoriels dans la categorie des k-algebres de Hopf
pointees, qui a tout espace vectoriel V associe son algebre de Hopf
bitensorielle pointee A(V). Cette algebre de Hopf est graduee, verifie une
propriete universelle, et contient une famille remarquable d'elements primitifs
P. Nous conjecturons que P engendre l'algebre de Lie des elements primitifs de
A(V). Enfin lorsque V est de dimension finie nous mettons en evidence un
couplage de Hopf entre A(V) et A(V*) dont le noyau contient l'ideal (de Hopf)
engendre par les elements de P de degre au moins egal a 2.


title: Bispectral Darboux Transformations: The Generalized Airy Case

abstract:
  This paper considers Darboux transformations of a bispectral operator which
preserve its bispectrality. A sufficient condition for this to occur is given,
and applied to the case of generalized Airy operators of arbitrary order $r>1$.
As a result, the bispectrality of a large family of algebras of rank $r$ is
demonstrated. An involution on these algebras is exhibited which exchanges the
role of spatial and spectral parameters, generalizing Wilson's rank one
bispectral involution. Spectral geometry and the relationship to the Sato
grassmannian are discussed.


title: Braided modules and reflection equations

abstract:
  We introduce a representation theory of q-Lie algebras defined earlier in
\cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss
other ways to define Lie algebra-like objects related to quantum groups, in
particular, those based on the so-called reflection equations. We also
investigate the truncated tensor product of braided modules.


title: Generalization and Exact Deformations of Quantum Groups

abstract:
  A large family of "standard" coboundary Hopf algebras is investigated. The
existence of a universal R-matrix is demonstrated for the case when the
parameters are in general position. Algebraic surfaces in parameter space are
characterized by the appearance of certain ideals; in this case the universal
R-matrix exists on the associated algebraic quotient. In special cases the
quotient is a "standard" quantum group; all familiar quantum groups including
twisted ones are obtained in this way. In other special cases one finds new
types of coboundary bi-algebras. The "standard" universal R-matrix is shown to
be the unique solution of a very simple, linear recursion relation. The
classical limit is obtained in the case of quantized Kac-Moody algebras of
finite and affine type. Returning to the general case, we study deformations of
the standard R-matrix and the associated Hopf algebras. A preliminary
investigation of the first order deformations uncovers a class of deformations
that incompasses the quantization of all Kac-Moody algebras of finite and
affine type. The corresponding exact deformations are described as generalized
twists, $ R_\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and
the cocycle $F$ (a power series in the deformation parameter $\epsilon$) is the
solution of a linear recursion relation of the same type as that which
determines $R$. Included here is the universal R-matrix for the elliptic
quantum groups associated with $sl(n)$, a big surprise! Specializing again, to
the case of quantized Kac-Moody algebras, and taking the classical limit of
these esoteric quantum groups, one re-discovers all the trigonometric and
elliptic r-matrices of Belavin and Drinfeld. The formulas obtained here are
easier to use than the original ones, and the structure of the space of
classical r-matrices is more transparent. The r-matrices obtained here are more
general in that they are defined on the full Kac-Moody algebras, the central
extensions of the loop groups.


title: On Associators and the Grothendieck-Teichmuller Group I

abstract:
  We present a formalism within which the relationship (discovered by
Drinfel'd) between associators (for quasi-triangular quasi-Hopf algebras) and
(a variant of) the Grothendieck-Teichmuller group becomes simple and natural,
leading to a simplification of Drinfel'd's original work. In particular, we
re-prove that rational associators exist and can be constructed iteratively,
though the proof itself still depends on the apriori knowledge that a
not-necessarily-rational associator exists.


title:
The bicovariant differential calculus on the kappa-Poincare and
  kappa-Weyl groups

abstract:
  The bicovariant differential calculus on fourdimensional kappa-Poincare group
and corresponding Lie-algebra like structure for any metric tensor are
described. The bicovariant differential calculus on four-dimensional kappa-Weyl
group and corresponding Lie-algebra like structure for any matric tensor in the
reference frame in which $g_{00}=0$ are considered.


title:
Geometric interpretation of the Poisson structure in affine Toda field
  theories

abstract:
  We express the Poisson brackets of local fields of the affine Toda field
theories in terms of the Drinfeld-Sokolov dressing operator. For this, we
introduce a larger space of fields, containing ``half screening charges'' and
``half integrals of motions''. In addition to local terms, the Poisson brackets
contain nonlocal terms related to trigonometric r-matrices.


title: Two-parameter quantum groups and quantum planes

abstract:
  Usually the generators of a quantum group are assumed to be commutative with
the noncommuting coordinates of a quantum plane. We have relaxed the assumption
and investigated its consequences. Not only does a two-parameter quantum group
arise naturally, but also the formulation leads us to many probable quantum
planes associated with a quantum group. Several examples are presented.


title: Polynomial Invariants are Polynomial

abstract:
  We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of
type $m$ is evaluated on a knot projection having $n$ crossings, the result is
bounded by a constant times $n^m$. Thus the well known analogy between
Vassiliev invariants and polynomials justifies (well, at least {\em explains})
the odd title of this note.


title:
Character States and Generator Matrix Elements for $Sp(4) \supset SU(2)
  \times U(1)$: Generic Representation

abstract:
  A new set of polynomial states (to be called character states) are derived
for $Sp(4)$ reduced to its $SU(2) \times U(1)$ subgroup, and the relevant
generator matrix elements are evaluated for generic representations $(a,b)$ of
$Sp(4)$. (The degenerate representations $(a,0)$ and $(0,b)$ were treated in
our previous work and are also given in this paper). The group--subgroup in
question is that of the seniority model of nuclear physics.


title: Vassiliev and Quantum Invariants of Braids

abstract:
  We prove that braid invariants coming from quantum gl(N) separate braids, by
recalling that these invariants (properly decomposed) are all Vassiliev
invariants, showing that all Vassiliev invariants of braids arise in this way,
and reproving that Vassiliev invariants separate braids. We discuss some
corollaries of this result and of our method of proof.


title: Quantum groups and q-lattices in phase space

abstract:
  Quantum groups lead to an algebraic structure that can be realized on quantum
spaces. These are noncommutative spaces that inherit a well defined
mathematical structure from the quantum group symmetry. In turn such quantum
spaces can be interpreted as noncommutative configuration spaces for physical
systems which carry a symmetry like structure. These configuration spaces will
be generalized to noncommutative phase space. The definition of the
noncommutative phase space will be based on a differential calculus on the
configuration space which is compatible with the symmetry. In addition a
conjugation operation will be defined which will allow us to define the phase
space variables in terms of algebraically selfadjoint operators. An interesting
property of the phase space observables will be that they will have a discrete
spectrum. These noncommutative phase space puts physics on a lattice structure.


title: Multivariable q-Racah polynomials

abstract:
  The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson
polynomials is studied for parameters satisfying a truncation condition such
that the orthogonality measure becomes discrete with support on a finite grid.
For this parameter regime the polynomials may be seen as a multivariable
counterpart of the (one-variable) $q$-Racah polynomials. We present the
discrete orthogonality measure, expressions for the normalization constants
converting the polynomials into an orthonormal system (in terms of the
normalization constant for the unit polynomial), and we discuss the limit $q\to
1$ leading to multivariable Racah type polynomials. Of special interest is the
situation that $q$ lies on the unit circle; in that case it is found that there
exists a natural parameter domain for which the discrete orthogonality measure
(which is complex in general) becomes real-valued and positive. We investigate
the properties of a finite-dimensional discrete integral transform for
functions over the grid, whose kernel is determined by the multivariable
$q$-Racah polynomials with parameters in this positivity domain.


title: Linear Connections on Graphs

abstract:
  In recent years, discrete spaces such as graphs attract much attention as
models for physical spacetime or as models for testing the spirit of
non-commutative geometry. In this work, we construct the differential algebras
for graphs by extending the work of Dimakis et al and discuss linear
connections and curvatures on graphs. Especially, we calculate connections and
curvatures explicitly for the general nonzero torsion case. There is a metric,
but no metric-compatible connection in general except the complete symmetric
graph with two vertices.


title: Canonical bases and the decomposition matrices of Ariki--Koike algebras

abstract:
  This paper has been withdrawn because of a gap in the proof of Lemma 3.10.
The main reults in this paper have now been proved, and extended in the
following papers:
  S. Ariki and A. Mathas, The number of simple modules of the Hecke algebras of
type G(r,1,n)
  S. Ariki, On the classification of simple modules for cyclotomic Hecke
algebras of type G(m,1,n) and Kleshchev multipartitions


title: On the Completeness of Some Subsystems of $q$-deformed Coherent States

abstract:
  The von Neumann type subsystems of $q$-deformed coherent states are
considered. The completeness of such subsystems is proved.


title: On quantum Jacobi identity

abstract:
  In this paper we present our variant of quantum antisymmetry and quantum
Jacobi identity.


title: Quantization of function algebras on semisimple orbits in $\g^*$

abstract:
  In this paper we describe a multiparameter deformation of the function
algebra of a semisimple coadjoint orbit. In the first section we use the
representation of the Lie algebra on a generalized Verma module to quantize the
Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit
type. In the second section we extend this construction to define a deformation
in the category of representations of the quantized enveloping algebra. In an
earlier paper we used cohomological methods to prove the existence of a two
parameter family quantizing a compatible pair of Poisson brackets on any
symmetric coadjoint orbit. This paper gives a more explicit algebraic
construction which includes more general orbit types and which we prove to be
flat in all parameters.


title: Null-plane Quantum Universal $R$-matrix

abstract:
  A non-linear map is applied onto the (non-standard) null-plane deformation of
(3+1) Poincar\'e algebra giving rise to a simpler form of this triangular
quantization. A universal $R$-matrix for the null plane quantum algebra is then
obtained from a universal $T$-matrix corresponding to a Hopf subalgebra.
Finally, the associated Poincar\'e Poisson--Lie group is quantized by using the
FRT approach.


title:
Convolutions for orthogonal polynomials from Lie and quantum algebra
  representations

abstract:
  The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials
as overlap coefficients in the positive discrete series representations of the
Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to
generalisations of the convolution identities for these polynomials. Using the
Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi
polynomials are obtained. From the quantised universal enveloping algebra for
su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the
Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah
coefficients. For the quantised universal enveloping algebra for su(2) q-Racah
polynomials are interpreted as Clebsch-Gordan coefficients, and the
linearisation coefficients for a two-parameter family of Askey-Wilson
polynomials are derived.


title: Demazure Modules and Perfect Crystals

abstract:
  We give a criterion for the Demazure crystal $B_w(\lambda)$ defined by
Kashiwara to have a tensor product structure. We study the $\sln$ symmetric
tensor case, and see some Demazure characters are expressed using
Kostka-Foulkes polynomials.


title: Differential Calculus on a three-parameter oscillator algebra

abstract:
  Two differential calculi are developped on an algebra generalizing the usual
q-oscillator algebra and involving three generators and three parameters. They
are shown to be invariant under the same quantum group that is extended to a
ten-generator Hopf algebra. We discuss the special case where it reduces to a
deformation of the invariance group of the Weyl-Heisenberg algebra for which we
prove the existence of a constraint between the values of the parameters.


title: Associative subalgebras of the Griess algebra and related topics

abstract:
  It is shown how certain idempotents in the Griess algebra generate the
discrete series representations for the Virasoro algebra inside the
Frenkel-Lepowsky-Meurman's moonshine module vertex operator algebra. It is also
shown that each Niemeier lattice determines (in many ways) certain maximal
associative subalgebras of the Griess algebra.


title: A Higher-Level Bailey Lemma: Proof and Application

abstract:
  In a recent letter, new representations were proposed for the pair of
sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma.
Here we extend and prove this result, providing pairs ($\gamma,\delta$)
labelled by the Lie algebra A$_{N-1}$, two non-negative integers $\ell$ and $k$
and a partition $\lambda$, whose parts do not exceed $N-1$. Our results give
rise to what we call a ``higher-level'' Bailey lemma. As an application it is
shown how this lemma can be applied to yield general $q$-series identities,
which generalize some well-known results of Andrews and Bressoud.


title: Flag varieties and the Yang-Baxter equation

abstract:
  We investigate certain bases of Hecke algebras defined by means of the
Yang-Baxter equation, which we call Yang-Baxter bases. These bases are
essentially self-adjoint with respect to a canonical bilinear form. In the case
of the degenerate Hecke algebra, we identify the coefficients in the expansion
of the Yang-Baxter basis on the usual basis of the algebra with specializations
of double Schubert polynomials. We also describe the expansions associated to
other specializations of the generic Hecke algebra.


title: Noncommutative Lattices and the Algebra of their Continuous Functions

abstract:
  Recently a new kind of approximation to continuum topological spaces has been
introduced, the approximating spaces being partially ordered sets (posets) with
a finite or at most a countable number of points. The partial order endows a
poset with a nontrivial non-Hausdorff topology. Their ability to reproduce
important topological information of the continuum has been the main motivation
for their use in quantum physics. Posets are truly noncommutative spaces, or
{\it noncommutative lattices}, since they can be realized as structure spaces
of noncommutative $C^*$-algebras. These noncommutative algebras play the same
role of the algebra of continuous functions ${\cal C}(M)$ on a Hausdorff
topological space $M$ and can be thought of as algebras of operator valued
functions on posets. In this article, we will review some mathematical results
that establish a duality between finite posets and a certain class of
C$^*$-algebras. We will see that the algebras in question are all postliminal
approximately finite dimensional (AF) algebras.


title: K-Theory of Noncommutative Lattices

abstract:
  Noncommutative lattices have been recently used as finite topological
approximations in quantum physical models. As a first step in the construction
of bundles and characteristic classes over such noncommutative spaces, we shall
study their $K$-theory. We shall do it algebraically, by studying the algebraic
$K$-theory of the associated algebras of `continuous functions' which turn out
to be noncommutative approximately finite dimensional (AF) $C^*$-algebra . We
also work out several examples.


title: Developing Computer Programs for Knot Classification

abstract:
  In this paper we summarise the work discussed in Ref. [1] and [2]
(q-alg/9505003), in which we introduced a method helpful in solving the problem
of knot classification. We also present results obtained since then.


title: Formal GNS Construction and States in Deformation Quantization

abstract:
  In this paper we develop a method of constructing Hilbert spaces and the
representation of the formal algebra of quantum observables in deformation
quantization which is an analog of the well-known GNS construction for complex
$C^*$-algebras: in this approach the corresponding positive linear functionals
(`states') take their values not in the field of complex numbers, but in (a
suitable extension field of) the field of formal complex Laurent series in the
formal parameter. By using the algebraic and topological properties of these
fields we prove that this construction makes sense and show in physical
examples that standard representations as the Bargmann and Schr\"odinger
representation come out correctly, both formally and in a suitable convergence
scheme. For certain Hamiltonian functions (contained in the Gel'fand ideal of
the positive functional) a formal solution to the time-dependent Schr\"odinger
equation is shown to exist. Moreover, we show that for every K\"ahler manifold
equipped with the Fedosov star product of Wick type all the classical delta
functionals are positive and give rise to some formal Bargmann representation
of the deformed algebra.


title:
Rogers-Schur-Ramanujan type identities for the $M(p,p')$ minimal models
  of conformal field theory

abstract:
  We present and prove Rogers-Schur-Ramanujan (Bose/Fermi) type identities for
the Virasoro characters of the minimal model $M(p,p').$ The proof uses the
continued fraction decomposition of $p'/p$ introduced by Takahashi and Suzuki
for the study of the Bethe's Ansatz equations of the XXZ model and gives a
general method to construct polynomial generalizations of the fermionic form of
the characters which satisfy the same recursion relations as the bosonic
polynomials of Forrester and Baxter. We use this method to get fermionic
representations of the characters $\chi_{r,s}^{(p,p')}$ for many classes of $r$
and $s.$


title: Rigidity of K-theory under deformation quantization

abstract:
  Quantization, at least in some formulations, involves replacing some algebra
of observables by a (more non-commutative) deformed algebra. In view of the
fundamental role played by K-theory in non-commutative geometry and topology,
it is of interest to ask to what extent K-theory remains "rigid" under this
process. We show that some positive results can be obtained using ideas of
Gabber, Gillet-Thomason, and Suslin. From this we derive that the algebraic
K-theory with finite coefficients of a deformation quantization of the
functions on a compact symplectic manifold, forgetting the topology, recovers
the topological K-theory of the manifold.


title: Habilitationsschrift: Renormalization and Knot Theory

abstract:
  We investigate to what extent renormalization can be understood as an
algebraic manipulation on concatenated one-loop integrals. We find that the
resulting algebra indicates a useful connection to knot theory as well as
number theory and report on recent results in support of this connection.


title: Anyonic Realization of the Quantum Affine Lie Algebras

abstract:
  We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in
terms of anyons defined on a one-dimensional chain (or on a two-dimensional
lattice), the deformation parameter $q$ being related to the statistical
parameter $\nu$ of the anyons by $q = e^{i\pi\nu}$. In the limit of the
deformation parameter going to one we recover the Feingold-Frenkel fermionic
construction of undeformed affine Lie algebras.


title: Creation operators for the Macdonald and Jack polynomials

abstract:
  Formulas of Rodrigues-type for the Macdonald polynomials are presented. They
involve creation operators, certain properties of which are proved and other
conjectured. The limiting case of the Jack polynomials is discussed.


title: Rodrigues formulas for the Macdonald polynomials

abstract:
  We present formulas of Rodrigues type giving the Macdonald polynomials for
arbitrary partitions through the repeated application of creation operators on
the constant 1. Three expressions for the creation operators are derived one
from the other. When the last of these expressions is used, the associated
Rodrigues formula readily implies the integrality of the (q,t)-Kostka
coefficients. The proofs given in this paper rely on the connection between
affine Hecke algebras and Macdonald polynomials


title:
A short proof of the integrality of the Macdonald (q,t)-Kostka
  coefficients

abstract:
  The Macdonald polynomials can be obtained by acting on the constant 1 with
creation operators. Three different expressions for these operators are
derived, one from the other, in a rather succint way. When the last of these
expressions is used, the formalism is seen to imply straightforwardly the
integrality of the (q,t)-Kostka coefficients, that is of the expansion
coefficients for the Macdonald functions in terms of Schur functions.


title:
Skew Young diagram method in spectral decomposition of integrable
  lattice models

abstract:
  The spectral decomposition of the path space of the vertex model associated
to the vector representation of the quantized affine algebra $U_q(\hat{sl}_n)$
is studied. We give a one-to-one correspondence between the spin configurations
and the semi-standard tableaux of skew Young diagrams. As a result we obtain a
formula of the characters for the degeneracy of the spectrum in terms of skew
Schur functions. We conjecture that our result describes the $sl_n$-module
contents of the Yangian $Y(sl_n)$-module structures of the level 1 integrable
modules of the affine Lie algebra $\hat{sl}_n$. An analogous result is obtained
also for a vertex model associated to the quantized twisted affine algebra
$U_q(A^{(2)}_{2n})$, where $Y(B_n)$ characters appear for the degeneracy of the
spectrum. The relation to the spectrum of the Haldane-Shastry and the
Polychronakos models are also discussed.


title: Examples of categorification

abstract:
  We construct tensor and bitensor categories with given Grothedieck rig
(fusion algebra) in simple cases. The results provide examples on which to test
the conjectural construction of 4-D TQFT's proposed by Crane and Frenkel and
shed light on several other constructions of TQFT's.


title: Gauge transformations and quasitriangularity

abstract:
  Natural conditions on a Poisson/quantum group G to implement Poisson/quantum
gauge transformations on the lattice are investigated. In addition to our
previous result that transformations on one lattice link require G to be
coboundary, it is shown that for a sequence of links one needs a
quasitriangular G.


title:
A decription of the quantum superalgebra $U_q[osp(2n+1/2m)]$ via Green
  generators

abstract:
  A description of the orthosymplectic Lie superalgebra $osp(2n+1/2m)$ and also
of its $q-$deformed analogue $U_q[osp(2n+1/2m)]$ in terms of a new set of
generators, called Green generators, is given. These generators are very
different form the well known Chevalley generators. Mathematically the Green
generators are the root vectors, corresponding to the positive and the negative
orthogonal roots. Physically, they consist of $m$ pairs of (deformed) para-Bose
operators and $n$ pairs of (deformed) para-Fermi operators.


title: Level-0 action of U_q(\hat{sl_n}) on the q-deformed Fock spaces

abstract:
  On the level-1 Fock space modules of the algebra $U_q(\hat{sl_n})$ we define
a level-0 action $U_0$ of the $U_q(\hat{sl_n})$, and an action of an abelian
algebra of conserved Hamiltonians commuting with the $U_0$. An irreducible
decomposition of the Fock space with respect to the level-0 action is derived
by constructing a base of the Fock space in terms of the Non-symmetric
Macdonald Polynomials.


title: On pentagon, ten-term, and tetrahedron relations

abstract:
  The tetrahedron equation in a special substitution is reduced to a pair of
pentagon and one ten-term equations. Various examples of solutions are found.
$O$-doubles of Novikov, which generalize the Heisenberg double of a Hopf
algebra, provide a particular algebraic solution to the problem.


title:
Zeros and poles of quantum current operators and the condition of
  quantum integrability

abstract:
  For the current realization of the affine quantum groups, a simple
comultiplication for the quantum current operators was given by Drinfeld. With
this comultiplication, we study the zeros and poles of the quantum current
operators and present a condition of integrability on the quantum current of
$U_q\left(\hat{\frak sl}(2)\right)$, which is a deformation of the
corresponding condition for $\hat{\frak sl}(2)$. We also present the results
about the zeros and poles of the quantum current operators of
$U_q\left(\hat{\frak sl}(n)\right)$.


title: Generalization and Deformation of Drinfeld quantum affine algebras

abstract:
  Drinfeld gave a current realization of the quantum affine algebras as a Hopf
algebra with a simple comultiplication for the quantum current operators. In
this paper, we will present a generalization of such a realization of quantum
Hopf algebras. As a special case, we will choose the structure functions for
this algebra to be elliptic functions to derive certain elliptic quantum groups
as a Hopf algebra, which degenerates into quantum affine algebras if we take
certain degeneration of the structure functions.


title: Drinfeld comultiplication and vertex operators

abstract:
  For the current realization of the quantum affine algebras, Drinfeld gave a
simple comultiplication of the quantum current operators. With this
comultiplication, we study the related vertex operators for the case of
$U_q(\hgtsl_n)$ and give an explicit bosonization of these new vertex
operators. We use these vertex operators to construct the quantum current
operators of $U_q(\hgtsl_n)$ and discuss its connection with quantum
boson-fermion correspondence.


title: Identities in Representations of the Hecke Algebra

abstract:
  New identities on traces of representations of the Hecke algebra on the
spaces of paths on graphs are presented. These identities are relevant in the
computation of partition functions with fixed boundary conditions and of
two-point correlation functions in a class of lattice models. They might also
help to determine the Boltzmann weights of these (allegedly) integrable models.


title: Quasi-Hopf algebras associated with sl(2) and complex curves

abstract:
  We construct quasi-Hopf algebras quantizing double extensions of the Manin
pairs of Drinfeld, associated to a curve with a meromorphic differential, and
the Lie algebra sl(2). This construction makes use of an analysis of the vertex
relations for the quantum groups obtained in our earlier work, PBW-type results
and computation of $R$-matrices for them; its key step is a factorization of
the twist operator relating ``conjugated'' versions of these quantum groups.


title: Fedosov *-products and quantum momentum maps

abstract:
  We study various aspects of Fedosov star-products on symplectic manifolds. By
introducing the notion of "quantum exponential maps", we give a criterion
characterizing Fedosov connections. As a consequence, a geometric realization
is obtained for the equivalence between an arbitrary *-product and a Fedosov
one. Every Fedosov *-product is shown to be a Vey *-product. Consequently, one
obtains that every *-product is equivalent to a Vey * -product, a classical
result of Lichnerowicz. Quantization of a hamiltonian G-space, and in
particular, quantum momentum maps are studied. Lagrangian submanifolds are also
studied under a deformation quantization.


title: Ohtsuki's Invariants are of Finite Type

abstract:
  Using a vanishing condition on certain combinations of components of the
Jones polynomial for algebraically split links we show that Ohtsuki's
invariants of integral homology three spheres are of finite type. We further
show that the corresponding manifold weight system is given by the expected Lie
algebraic construction.


title: Special functions and q-commuting variables

abstract:
  This paper is mostly a survey, with a few new results. The first part deals
with functional equations for q-exponentials, q-binomials and q-logarithms in
q-commuting variables and more generally under q-Heisenberg relations. The
second part discusses translation invariance of Jackson integrals, q-Fourier
transforms and the braided line.


title: Compact automorphism groups of vertex operator algebras

abstract:
  Let $V$ be a simple vertex operator algebra which admits the continuous,
faithful action of a compact Lie group $G$ of automorphisms. We establish a
Schur-Weyl type duality between the unitary, irreducible modules for $G$ and
the irreducible modules for $V^G$ which are contained in $V$ where $V^G$ is the
space of $G$-invariants of $V.$ We also prove a concomitant Galois
correspondence between vertex operator subalgebras of $V$ which contain $V^G$
and closed Lie subgroups of $G$ in the case that $G$ is abelian.


title: The Problem of Differential Calculus on Quantum Groups

abstract:
  The bicovariant differential calculi on quantum groups of Woronowicz have the
drawback that their dimensions do not agree with that of the corresponding
classical calculus. In this paper we discuss the first-order differential
calculus which arises from a simple quantum Lie algebra. This calculus has the
correct dimension and is shown to be bicovariant and complete. But it does not
satisfy the Leibniz rule. For sl_n this approach leads to a differential
calculus which satisfies a simple generalization of the Leibniz rule.


title: New rational solutions of Yang-Baxter equation and deformed Yangians

abstract:
  In this paper a class of new quantum groups is presented: deformed Yangians.
They arise from rational solutions of the classical Yang-Baxter equation of the
form $c_2 /u + const$ . The universal quantum $R$-matrix for a deformed Yangian
is described. Its image in finite-dimensional representaions of the Yangian
gives new matrix rational solutions of the Yang-Baxter equation (YBE).


title: Dirac operators on quantum $SU(2)$ group and quantum sphere

abstract:
  Definition of Dirac operators on the quantum group $SU_{q}(2)$ and the
quantum sphere $S^{2}_{q \mu}$ are discussed. In both cases similar
$SU_{q}(2)$-invariant form is obtained. It is connected with corresponding
Laplace operators.


title: A new scalar product for nonsymmetric Jack polynomials

abstract:
  Symmetric Jack polynomials arise naturally in several contexts, including
statistics, physics, combinatorics, and representation theory. They are
pairwise orthogonal with repsect to two different inner products, the first
defined by integration over an n-dimensional torus, and the second defined via
a power series expansion.
  The nonsymmetric analogs of Jack polynomials were recently defined by Opdam
via the first inner product. In this paper we show that they are also
orthogonal with respect to an analog of the second product which was proposed
by Dunkl. We also derive an explict formula for their norms.


title: Spin Topological Quantum Field Theories

abstract:
  Starting from the quantum group SL_q(2,C), we construct operator invariants
of 3-cobordisms with spin structure, satisfying the requirements of a
topological quantum field theory and refining the Reshetikhin--Turaev and
Turaev--Viro models. We establish the relationship between these two refined
models.


title: On the Two q-Analogue Logarithmic Functions

abstract:
  There is a simple, multi-sheet Riemann surface associated with e_q(z)'s
inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be
defined. However, the topology of the Riemann surface for ln_q(w) changes each
time "q" increases above the collision point of a pair of the turning points of
e_q(x). There is also a power series representation for ln_q(1+w). An
infinite-product representation for e_q(z) is used to obtain the ordinary
natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of
e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power
series in terms of values of these sum rules. The values of the sum rules for
the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the
usual Bernoulli numbers.


title: Classification of Bicovariant Differential Calculi

abstract:
  We show that the bicovariant first order differential calculi on a
factorisable semisimple quantum group are in 1-1 correspondence with
irreducible representations $V$ of the quantum group enveloping algebra. The
corresponding calculus is constructed and has dimension $dim V^2$. The
differential calculi on a finite group algebra $C G$ are also classified and
shown to be in correspondence with pairs consisting of an irreducible
representation $V$ and a continuous parameter in $C P^{dim V -1}$. They have
dimension $dim V$. For a classical Lie group we obtain an infinite family of
non-standard calculi. General constructions for bicovariant calculi and their
quantum tangent spaces are also obtained.


title: Symmetric Multiplets in Quantum Algebras

abstract:
  We consider a modified version of the coproduct for $\U(\su_q(2))$ and show
that in the limit when $q \rightarrow 1$, there exists an essentially
non-cocommutative coproduct. We study the implications of this
non-cocommutativity for a system of two spin-$1/2$ particles. Here it is shown
that, unlike the usual case, this non-trivial coproduct allows for symmetric
and anti-symmetric states to be present in the multiplet. We surmise that our
analysis could be related to the ferromagnetic and antiferromagnetic cases of
the Heisenberg magnets.


title: 8 Lectures on quantum groups and q-special functions

abstract:
  Lecture notes for an eight hour course on quantum groups and $q$-special
functions at the fourth Summer School in Differential Equations and Related
Areas, Universidad Nacional de Colombia and Universidad de los Andes, Bogot\'a,
Colombia, July 22 -- August 2, 1996. The lecture notes contain an introduction
to quantum groups, $q$-special functions and their interplay. After
generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric
series we work out the relation between the quantum SU(2) group and the
Askey-Wilson polynomials out in detail as the main example. As an application
we derive an addition formula for a two-parameter subfamily of Askey-Wilson
polynomials. A relation between the Al-Salam and Chihara polynomials and the
quantised universal enveloping algebra for $su(1,1)$ is given. Finally, more
examples and other approaches as well as some open problems are given.


title:
Bicovariant Differential Calculi and Cross Products on Braided Hopf
  Algebras

abstract:
  We consider Hopf bimodules and crossed modules over a Hopf algebra $H$ in a
braided category. They are the key-stones for braided bicovariant differential
calculi and their invariant vector fields respectively, as well as for the
construction of braided Hopf algebra cross products. We show that the notions
of Hopf bimodules and crossed modules are equivalent. A generalization of the
Radford-Majid criterion to the braided case is given and it is seen that
bialgebra cross products over the Hopf algebra $H$ are precisely described by
$H$-crossed module bialgebras. We study the theory of (bicovariant)
differential calculi in braided abelian categories and we construct
$\NN_0$-graded bicovariant differential calculi out of first order bicovariant
differential calculi. These objects are shown to be Hopf algebra differential
calculi with universal bialgebra properties in the braided $\NN_0$-graded
category.


title: Shifted Jack polynomials, binomial formula, and applications

abstract:
  In this note we prove an explicit binomial formula for Jack polynomials and
discuss some applications of it.


title: Binomial formula for Macdonald polynomials

abstract:
  We prove a binomial formula for Macdonald polynomials and consider
applications of it.


title: The radical of a vertex operator algebra

abstract:
  The radical $J(V)$ of a vertex operator algebra $V$ is defined to be the
subspace of $V$ consisting of vectors $v$ such that the zero mode $o(v)=0$ on
$V$ where $o(v)=v_{wt v-1}$ if $v$ is homogeneous. We establish various facts
about $o(v),$ including the determination of $J(V)$ which is shown to be
essentially equal to $(L(0)+L(-1))V.$