title: Unified view of multimode algebras with Fock-like representations

abstract:
  A unified view of general multimode oscillator algebras with Fock-like
representations is presented.It extends a previous analysis of the single-mode
oscillator algebras.The expansion of the $a_ia_j^{\dagger}$ operators is
extended to include all normally ordered terms in creation and annihilation
operators and we analyze their action on Fock-like states.We restrict ourselves
to the algebras compatible with number operators. The connection between these
algebras and generalized statistics is analyzed.We demonstrate our approach by
considering the algebras obtainable from the generalized Jordan-Wigner
transformation, the para-Bose and para-Fermi algebras, the Govorkov
"paraquantization" algebra and generalized quon algebra.


title:
Crystallized Peter-Weyl type decomposition for level 0 part of modified
  quantum algebra $\widetilde U_q(\widehat{sl_2})_0$

abstract:
  We shall give some criteria for the existence of the Peter-Weyl type
decomposition for modified quantum algebra, which is motivated by the fact that
modified quantum algebra has commutative two crystal structures. We shall apply
the criteria to the crystal base of the level 0 part of the modified quantum
affine algebra $\widetilde U_q(\widehat{sl_2})_0$ and give more precise
description of Peter-Weyl type decomposition.


title: The Einstein Action for Algebras of Matrix Valued Functions - Toy Models

abstract:
  Two toy models are considered within the framework of noncommutative
differential geometry. In the first one, the Einstein action of the Levi-Civita
connection is computed for the algebra of matrix valued functions on a torus.
It is shown that, assuming some constraints on the metric, this action splits
into a classical-like, a quantum-like and a mixed term. In the second model, an
analogue of the Palatini method of variation is applied to obtain critical
points of the Einstein action functional for $M\sb 4(R)$. It is pointed out
that a solution to the Palatini variational problem is not necessarily a
Levi-Civita connection. In this model, no additional assumptions regarding
metrics are made.


title: Vassiliev invariants I: Braid groups and rational homotopy theory

abstract:
  We get a detailed account of Vassiliev type invariants starting with Chen's
theory of iterated integrals and Malcev's completion of discrete groups. The
canonical injection of the group of pure braids into its completion is
identified with the universal Kontsevich-Vassiliev invariant.Further we discuss
the extension of this morphism to the whole braid group and the multiplication
law for the last one.


title: Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories

abstract:
  Hopf (bi-)modules and crossed modules over a bialgebra B in a braided
monoidal category C are considered. The (braided) monoidal equivalence of both
categories is proved provided B is a Hopf algebra (with invertible antipode).
Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are
found to be isomorphic categories. As a consequence a generalization of the
Radford-Majid criterion for a braided Hopf algebra to be a cross product is
obtained. The results of this paper turn out to be fundamental for the
construction of (bicovariant) differential calculi on braided Hopf algebras.


title: On Reduced Q-Functions

abstract:
  Schur's $Q$-functions with reduced variables are discussed by employing a
combinatorics of strict partitions. They are called reduced $Q$-functions. We
give a description of the linear relations among reduced $Q$-functions.


title: Deformed Minkowski spaces: clasification and properties

abstract:
  Using general but simple covariance arguments, we classify the `quantum'
Minkowski spaces for dimensionless deformation parameters. This requires a
previous analysis of the associated Lorentz groups, which reproduces a previous
classification by Woronowicz and Zakrzewski. As a consequence of the unified
analysis presented, we give the commutation properties, the deformed (and
central) length element and the metric tensor for the different spacetime
algebras.


title: On braided FRT-construction

abstract:
  Fully braided analog of Faddeev-Reshetikhin-Takhtajan construction of
quasitriangular bialgebra $A(X,R)$ is proposed. For given pairing $C$
factor-algebra $A(X,R;C)$ is a dual quantum braided group. Corresponding
inhomogeneous quantum group is obtained as a result of generalized
bosonization. Construction of first order bicovariant differential calculus is
proposed.


title: Crossed modules and quantum groups in braided categories

abstract:
  Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules
over $A$ are introduced and studied as objects with both module and comodule
structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$
of crossed modules is braided and is a concrete realization of a known general
construction of a double or center of a monoidal category. For a quantum
braided group $(A,\overline A,{\cal R})$ the corresponding braided category of
modules ${\cal C}_{\cO{A,\overline A}}$ is identified with a full subcategory
in $\DY{\cal C}_A^A$. The connection with cross products is discussed and a
suitable cross product in the class of quantum braided groups is built.
Majid--Radford theorem, which gives equivalent conditions for an ordinary Hopf
algebra to be such a cross product, is generalized to the braided category.
Majid's bosonization theorem is also generalized.


title:
Spinor Representations of $U_q(\hat{\frak gl}(n))$ and Quantum
  Boson-Fermion Correspondence

abstract:
  This is an extension of quantum spinor construction in \cite{DF2}. We define
quantum affine Clifford algebras based on the tensor category and the solutions
of q-KZ equations, construct quantum spinor representations of $U_q(\hat{\frak
gl}(n))$ and explain classical and quantum boson-fermion correspondence.


title: Generalised Magnus modules over the braid group

abstract:
  W.~Magnus' representations of submonoids $ E \leq \mbox{End}(F) $ of the
endomorphisms of a free group $ F $ of finite rank are generalised by
identifying them with the first homology group of $ F $ with particular
coefficient modules. By considering a suitable free resolution of the integers
over the semidirect product of free groups, a class of representations of the
braid group can be obtained on higher homology groups. The resolution shows
that the holonomy representations of the braid group and of the Hecke algebra
constructed topologically by R.~J.~Lawrence belong to this class.


title: The Alexander- and Jones-invariants and the Burau module

abstract:
  From the braid-valued Burau module over the braid group we construct the
Yang-Baxter matrices yielding the Alexander- and the Jones knot invariants.
This generalises an observation of V. F. R. Jones.


title:
Representation of $SU_q(2)$-covariant $q$-Lie Algebra in Terms of
  Differential Operators

abstract:
  A three-dimensional $q$-Lie algebra of $SU_q(2)$ is realized in terms of
first- and second-order differential operators. Starting from the $q$-Lie
algebra one has constructed a left-covariant differential calculus on the
quantum group. The proposed construction is inverse to the standard Woronowicz
approach; the left-invariant vector fields are introduced as initial objects
whereas the differential 1-forms are defined in a dual manner.


title:
Quasitriangularity and enveloping algebras for inhomogeneous quantum
  groups

abstract:
  Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and
quantum Poincar\'e groups are classified. The results extend (under certain
assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on
those objects are described.


title:
Solutions of Klein--Gordon and Dirac equations on quantum Minkowski
  spaces

abstract:
  Covariant differential calculi and exterior algebras on quantum homogeneous
spaces endowed with the action of inhomogeneous quantum groups are classified.
In the case of quantum Minkowski spaces they have the same dimensions as in the
classical case. Formal solutions of the corresponding Klein--Gordon and Dirac
equations are found. The Fock space construction is sketched.


title: Quantization of Poisson algebraic groups and Poisson homogeneous spaces

abstract:
  This paper consists of two parts. In the first part we show that any Poisson
algebraic group over a field of characteristic zero and any Poisson Lie group
admits a local quantization. This answers positively a question of Drinfeld. In
the second part we apply our techniques of quantization to obtain some
nontrivial examples of quantization of Poisson homogeneous spaces.


title: Geometry of the Quantum Complex Projective Space $CP_q(N)$

abstract:
  The quantum deformation $CP_q(N)$ of complex projective space is discussed.
Many of the features present in the case of the quantum sphere can be extended.
The differential and integral calculus is studied and $CP_q(N)$ appears as a
quantum K\"{a}hler manifold. The braiding of several copies of $CP_q(N)$ is
introduced and the anharmonic ratios of four collinear points are shown to be
invariant under quantum projective transformations. They provide the building
blocks of all projective invariants. The Poisson limit is also described.


title: The Use of Quantum Groups in Nuclear Structure Problems

abstract:
  Various applications of quantum algebraic techniques in nuclear structure
physics, such as the su$_q$(2) rotator model and its extensions, the use of
deformed bosons in the description of pairing correlations, and the
construction of deformed exactly soluble models (Interacting Boson Model,
Moszkowski model) are briefly reviewed. Emphasis is put in the study of the
symmetries of the anisotropic quantum harmonic oscillator with rational ratios
of frequencies, which underly the structure of superdeformed and hyperdeformed
nuclei, the Bloch--Brink $\alpha$-cluster model and possibly the shell
structure in deformed atomic clusters.


title:
Connection Between q-Deformed Anharmonic Oscillators and Quasi-Exactly
  Soluble Potentials

abstract:
  It is proved that quasi-exactly soluble potentials (QESPs) 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 all levels of the QESPs.


title: Nonsemisimple Quantum Groups as Hopf Algebras of the Dual Functions

abstract:
  The nonsemisimple quantum Cayley-Klein groups $ Fun(SU_{q}(2;\bf j}) $ are
realized as Hopf algebra of the noncommutative functions with the dual (or
Study) variables. The {\it dual} quantum algebras $ su_q(2;{\bf j}) $ are
constructed and their isomorphisms with the corresponding quantum orthogonal
algebras $ so_q(3;{\bf j}) $ are established. The possible couplings of the
Cayley-Klein and Hopf structures are considered.


title:
Differential Calculi on the Quantum Group $SU_q(2)$ and Global
  $U(1)$-covariance

abstract:
  A variety of three-dimensional left-covariant differential calculi on the
quantum group $SU_q(2)$ is considered using an approach based on global $ U(1)
$ -covariance. Explicit representations of possible $q $-Lie algebras are
constructed in terms of differential operators. A gauge covariant differential
algebra is uniquely determined. The non-standard Leibnitz rule is obtained for
a corresponding $q$-Lie algebra.


title:
Kac-Peterson, Perron-Frobenius, and the Classification of Conformal
  Field Theories

abstract:
  The classification of CFTs has an important subproblem, namely classifiying
the partition functions for WZW theories. This subproblem is intimately
connected to the modular behaviour of the characters of affine algebras. This
paper addresses the classification of the largest class of these -- the natural
generalization to any affine algebra of the A-series, D-series, and
$E_7$-exceptional appearing in the A-D-E classification of
Cappelli-Itzykson-Zuber for $A_1^{(1)}$. We give a programme for classifying
these for any algebra, and explicitly do this for the most interesting case:
$A_r^{(1)}$ at any level k. The exceptionals appear at (r,k)=(1,16), (2,9),
(3,8), (4,5), (7,4), (8,3) and (15,2).


title: Double-Bosonisation and the Construction of {$U_q(g)$}

abstract:
  We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the
{\em double-bosonisation}, associated to every braided group $B$ in the
category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$
appears as the `positive root space', $H$ as the `Cartan subalgebra' and the
dual braided group $B^*$ as the `negative root space' of $U(B)$. The choice
$B=f$ recovers Lusztig's construction of $U_q(g)$, where $f$ is Lusztig's
algebra associated to a Cartan datum; other choices give more novel quantum
groups. As an application, our construction provides a canonical way of
building up quantum groups from smaller ones by repeatedly extending their
positive and negative root spaces by linear braided groups; we explicitly
construct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the
quantum-braided plane $A_q^2$. We provide a fundamental representation of
$U(B)$ in $B$. A projection from the quantum double, a theory of double
biproducts and a Tannaka-Krein reconstruction point of view are also provided.


title: Phase spaces related to standard classical $r$-matrices

abstract:
  Fundamental representations of real simple Poisson Lie groups are Poisson
actions with a suitable choice of the Poisson structure on the underlying
(real) vector space. We study these (mostly quadratic) Poisson structures and
corresponding phase spaces (symplectic groupoids).


title: Deformations of the KdV hierarchy and related soliton equations

abstract:
  We define hierarchies of differential--q-difference equations, which are
q-deformations of the equations of the generalized KdV hierarchies. We show
that these hierarchies are bihamiltonian, one of the hamiltonian structures
being that of the q-deformed classical W-algebra of sl(N), defined by
Reshetikhin and the author. We also find q-deformations of the mKdV hierarchies
and the affine Toda equations.


title: Gauss Decomposition for Quantum Groups and Duality

abstract:
  The Gauss decomposition of quantum groups and supergroups are considered. The
main attention is paid to the R-matrix formulation of the Gauss decomposition
and its properties as well as its relation to the contraction procedure.
Duality aspects of the Gauss decomposition are also touched. For clarity of
exposition a few simple examples are considered in some details.


title: Deformation of Yangian $Y(sl_2)$

abstract:
  A quantization of a non-standard rational solution of CYBE for $sl_2$ is
given explicitly. We obtain the quantization with the help of a twisting of the
usual Yangian $Y(sl_2$. This quantum object (deformed Yangian
$Y_{\eta,\xi}(sl_2))$ is a two-parametric deformation of the universal
enveloping algebra $U(sl_2[u])$ of the positive current algebra $sl_2[u]$. We
consider the pseudotriangular structure on $Y_{\eta,\xi}(sl_2)$, the quantum
double $DY_{\eta,\xi}(sl_2)$ its the universal R-matrix and also the
RTT-realization of $Y_{\eta,\xi}(sl_2)$.


title: Twisted Yang-Baxter equations for linear quantum (super)groups

abstract:
  We consider the modified (or twisted) Yang-Baxter equations for the
$SL_{q}(N)$ groups and $SL_{q}(N|M)$ supergroups. The general solutions for
these equations are presented in the case of the linear quantum (super)groups.
The introduction of spectral parameters in the twisted Yang-Baxter equation and
its solutions are also discussed.


title: Intertwining Operators And Quantum Homogeneous Spaces

abstract:
  In the present paper the algebras of functions on quantum homogeneous spaces
are studied. The author introduces the algebras of kernels of intertwining
integral operators and constructs quantum analogues of the Poisson and Radon
transforms for some quantum homogeneous spaces. Some applications and the
relation to $q$-special functions are discussed.


title: Cycle for integration for zonal spherical function of type $A_n$

abstract:
  Integral of a certain multivalued form over cycle $\pmb\Delta$ provides zonal
spherical function of type $A_n$. This paper is devoted to quantum group
analysis and verification of monodromy properties of the distinguished cycle
$\pmb\Delta$. Zonal spherical function is a particular conformal block of
$WA_n$-algebra.


title: h-deformation of GL(1|1)

abstract:
  h-deformation of (graded) Hopf algebra of functions on supergroup GL(1|1) is
introduced via a contration of GL_q (1|1). The deformation parameter h is odd
(grassmann). Related differential calculus on h-superplane is presented.


title: A gauge-field approach to 3- and 4-manifold invariants

abstract:
  The paper contains a talk given by the author at the Banach Center in Spring
1995. It recapitulates author's approach to construction of topological
invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional
manifolds in the framework of SU(2) Chern-Simons gauge theory and its hidden
(quantum) gauge symmetry.


title:
Modular functor and representation theory of \hat{sl}_{2} at a rational
  level

abstract:
  We define a new modular functor based on Kac-Wakimoto admissible
representations and the corresponding D-module on the moduli space of rank 2
vector bundles with parabolic structure. A new fusion functor arises which is
related to representation theory of the pair "osp(1|2),sl(2)" in the same way
as the fusion functor for the Virasoro algebra is related to representation
theory of the pair "sl(2),sl(2)"


title: Isotopic pairs and their representations. II. A general supercase

abstract:
  Isotopic pairs and their representations are considered in a general
framework of the vector superalgebra. Numerous examples of finite-dimensional
and infinite-dimensional isotopic pairs are discussed. Several types of their
representations are analysed. Applications to physics and relations to other
mathematical objects are specified. Some curious combinatorial aspects of
representation theory of isotopic pairs (graph-representations) are explicated.
Perspectives of the further researches are drawn.


title: Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

abstract:
  We begin with a brief sketch of what is known and conjectured concerning
braided monoidal 2-categories and their applications to 4d topological quantum
field theories and 2-tangles (surfaces embedded in 4-dimensional space). Then
we give concise definitions of semistrict monoidal 2-categories and braided
monoidal 2-categories, and show how these may be unpacked to give long explicit
definitions similar to, but not quite the same as, those given by Kapranov and
Voevodsky. Finally, we describe how to construct a semistrict braided monoidal
2-category Z(C) as the `center' of a semistrict monoidal category C. This is
analogous to the construction of a braided monoidal category as the center, or
`quantum double', of a monoidal category. As a corollary, our construction
yields a strictification theorem for braided monoidal 2-categories.


title: Quantum hyperboloid and braided modules

abstract:
  When a quantum hyperboloid is realized, as a three - parameter algebra
$\ahqc$, in the usual manner, the following problem arises: what is a
``representation theory'' of this algebra? We construct the series of all spin
representations of $\ahqc$, and we discuss a braided version of the orbit
method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules.
A braided trace and a braided involution are discussed as well.


title: Quantum Dilogarithm and Tetrahedron Equation

abstract: This paper has been temporarily withdrawn for corrections.


title: Solution of Constant Yang-Baxter System in the Dimension Two

abstract:
  Complete solution, more precisely, all invertible $4\times 4$ matrices $R,Q$
that solve Yang--Baxter system related to quantised braided groups, quantum
doubles and other systems are given.


title:
Multiparameter Deformations of the Algebra $gl_n$ in Terms of Anyonic
  Oscillators

abstract:
  Generators of multiparameter deformations $U_{q;s_1,s_2,...,s_{n-1}}(gl_n)$
of the universal enveloping algebra $U(gl_n)$ are realized bilinearly by means
of appropriately generalized form of anyonic oscillators (AOs). This
modification takes into account the parameters $s_1,...,s_{n-1}$ and yields
usual AOs when all the $s_i$ are set equal to unity.


title: Unitary operator bases and q-deformed algebras

abstract:
  Starting from the Schwinger unitary operator bases formalism constructed out
of a finite dimensional state space, the well-known q-deformed commutation
relation is shown to emerge in a natural way, when the deformation parameter is
a root of unity.


title:
A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and
  difference equations

abstract:
  We consider the universal solution of the Gervais-Neveu-Felder equation in
the ${\cal U}_q(sl_2)$ case. We show that it has a quasi-Hopf algebra
interpretation. We also recall its relation to quantum 3-j and 6-j symbols.
Finally, we use this solution to build a q-deformation of the trigonometric
Lam\'e equation.


title: Dilute Algebras and Solvable Lattice Models

abstract:
  The definition of a dilute braid-monoid algebra is briefly reviewed. The
construction of solvable vertex and interaction-round-a-face models built on
representations of the dilute Temperley-Lieb and Birman-Wenzl-Murakami algebras
is discussed.


title: Koszul Property and Poincare' Series of Matrix-Bialgebras of Type $A_N$

abstract:
  Bialgebras associated to Yang-Baxter operators satisfying the Hecke equation,
are considered. It is shown that they are Koszul algebras. Their Poincare'
series are calculated via the Poincare' series of the corresponding quantum
spaces.


title: Quantization of Poisson Groups

abstract:
  Let $ G^\tau $ be a connected simply connected semisimple algebraic group,
endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $
H^\tau $ be its dual Poisson group. By means of quantum double construction and
dualization via formal Hopf algebras, we construct new quantum groups $
U_{q,\varphi}^M(\frak{h}) $ --- dual of $ U_{q,\varphi}^{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: On The Structure of Quantum Super Groups GL$_q(m|n)$

abstract:
  We show that a quantum super matrix in standard format is invertible if and
only if its block matrices of even entries are invertible. We prove the
q-analog of the well-known formula for the Berezinian.


title: Cabling the Vassiliev Invariants

abstract:
  We characterise the cabling operations on the weight systems of finite type
knot invariants. The eigenvectors and eigenvalues of this family of operations
are described over the cable eigenbasis. The action of immanent weight systems
on general Feynman diagrams is considered, and the highest eigenvalue cabling
eigenvectors are shown to be dual to the immanent weight systems. Using these
results, we prove a recent conjecture of Bar-Natan and Garoufalidis on cablings
of weight systems.


title:
On Finite Type Invariants of Links and Rational Homology Spheres Derived
  from the Jones Polynomial and Witten-Reshetikhin-Turaev Invariant

abstract:
  We present a mathematically clean review of our previous results on 1/K
expansion of the colored Jones polynomial and on perturbative invariants of 3d
rational homology spheres. We also prove that perturbative invariants defined
through the stationary phase surgery formula are invariant under Kirby moves.


title: Twining characters, orbit Lie algebras, and fixed point resolution

abstract:
  We describe the resolution of field identification fixed points in coset
conformal field theories in terms of representation spaces of the coset chiral
algebra. A necessary ingredient from the representation theory of Kac Moody
algebras is the recently developed theory of twining characters and orbit Lie
algebras, as applied to automorphisms representing identification currents.


title: Some aspects of noncommutative differential geometry

abstract:
  We discuss in some generality aspects of noncommutative differential geometry
associated with reality conditions and with differential calculi. We then
describe the differential calculus based on derivations as generalization of
vector fields, and we show its relations with quantum mechanics. Finally we
formulate a general theory of connections in this framework.


title:
Multiparametric and coloured extensions of the quantum group $GL_q(N)$
  and the Yangian algebra $Y(gl_N)$ through a symmetry transformation of the
  Yang-Baxter equation

abstract:
  Inspired by Reshetikhin's twisting procedure to obtain multiparametric
extensions of a Hopf algebra, a general `symmetry transformation' of the
`particle conserving' $R$-matrix is found such that the resulting
multiparametric $R$-matrix, with a spectral parameter as well as a colour
parameter, is also a solution of the Yang-Baxter equation (YBE). The
corresponding transformation of the quantum YBE reveals a new relation between
the associated quantized algebra and its multiparametric deformation. As
applications of this general relation to some particular cases, multiparametric
and coloured extensions of the quantum group $GL_q(N)$ and the Yangian algebra
$Y(gl_N)$ are investigated and their explicit realizations are also discussed.
Possible interesting physical applications of such extended Yangian algebras
are indicated.


title:
Levi-Civita Connections on the Quantum Groups SL_{q}(N), O_{q}(N) and
  Sp_{q}(N)

abstract:
  For bicovariant differential calculi on quantum groups various notions on
connections and metrics (bicovariant connections, invariant metrics, the
compatibility of a connection with a metric, Levi-Civita connections) are
introduced and studied. It is proved that for the bicovariant differential
calculi on $SL_{q}(N)$, $O_{q}(N)$ and $Sp_{q}(N)$ from the classification of
Schmuedgen, K. and Schueler, A. there exist unique Levi-Civita connections.


title: On a Universal Invariant of 3-Manifolds

abstract:
  We construct an invariant of 3-manifolds using a modification of the
Kontsevich integral and Kirby's calculus. This invariant, as expected in
perturbative Chern-Simon theory, takes values in the algebra of oriented
3-valent graphs. This algebra is a Hopf algebra, graded by half the number of
vertices in 3-valent graphs. The degree 1 term of the invariant coincides with
Casson-Walker-Lescop invariant. The degree $n$ term is constructed out of the
universal Vassiliev invariant of links of degree less than or equal to $(l+1)n$
where $l$ is the number of link components.


title: A_2 Macdonald polynomials: a separation of variables

abstract:
  In this paper we construct a discrete linear operator $K$ which transforms
$A_2$ Macdonald polynomials into the product of two basic $3\phi_2$
hypergeometric series with known arguments. The action of the operator $K$ on
power sums in two variables can be reduced to a generalization of one
particular case of the Bailey's summation formula for a very-well-poised
$6\psi_6$ series. We also propose the conjecture for a transformation of
$6\psi_6$ series with different arguments.


title: On Curvature in Noncommutative Geometry

abstract:
  A general definition of a bimodule connection in noncommutative geometry has
been recently proposed. For a given algebra this definition is compared with
the ordinary definition of a connection on a left module over the associated
enveloping algebra. The corresponding curvatures are also compared.


title: Vertex Operator of $U_q(\widehat{B_l})$ for Level One

abstract:
  In this paper,we give the explicit formulae of vertex operators of
${U_q}(\widehat{B}_l)$ for level-one as operators on the Fock space. Meanwhile,
we point out that the free field realization (by one fermionic field and $l$
bosonic fields) of highest weight module with highest weight $\Lambda_l$ has
two irreducible modules.


title: Fermionic $q$-Fock Space and Braided Geometry

abstract:
  We write the fermionic $q$-Fock space representation of $U_q(\hat{sl_n})$ as
an infinite extended braided tensor product of finite-dimensional fermionic
$U_q(sl_n)$-quantum planes or exterior algebras. Using braided geometrical
techniques developed for such quantum exterior algebras, we provide a new
approach to the Kashiwara-Miwa-Stern action of the Heisenberg algebra on the
$q$-fermionic Fock space, obtaining the action in detail for the lowest
nontrivial case $[b_{2},b_{-2}]=2({1-q^{-4n}\over 1-q^{-4}})$. Our R-matrix
approach includes other Hecke R-matrices as well.


title: Bicrossproduct Structure of Affine Quantum Groups

abstract:
  We show that the affine quantum group $U_q(\hat{sl_2})$ is isomorphic to a
bicrossproduct central extension $C Z{}_\chi\cross U_q(Lsl_2)$ of the quantum
loop group $U_q(Lsl_2)$ by a quantum cocycle $\chi$, which we construct. We
prove the same result for $U_q(\hat{g})$ in R-matrix form.


title: Braided antisymmetrizer as bialgebra homomorphism

abstract:
  For an Yang Baxter operator we show that a bialgebra homomorphism from a free
braided tensor bialgebra to a cofree braided shuffle bialgebra is the
Woronowicz braided antisymmetrizer. A cofree braided shuffle bialgebra is a
braided generalization of a cofree shuffle bialgebra introduced by Sweedler.
Its graded dual bialgebra is a free braided tensor bialgebra.


title: Koszul Theorem for S-Lie coalgebras

abstract:
  For a symmetry braid S-Lie coalgebras, as a dual object to algebras
introduced by Gurevich, are considered. For an Young antisymmetrizer an
S-exterior algebra is introduced. From this differential point of view S-Lie
coalgebras are investigated. The dual Koszul theorem in this case is proved.


title: Kadell's two conjectures for Macdonald polynomials

abstract:
  This paper contains the proof of difference counterparts of the conjectures
due to Keven Kadell on symmetric and anti-symmetric Macdonald polynomials.


title: Compatible Poisson-Lie structures on the loop group of $SL_{2}$

abstract:
  We define a 1-parameter family of $r$-matrices on the loop algebra of
$sl_{2}$, defining compatible Poisson structures on the associated loop group,
which degenerate into the rational and trigonometric structures, and study the
Manin triples associated to them.


title: Geometrical Quantization in Fock Space

abstract:
  We investigate an infinite dimensional analog of the theory of Lagrangian
manifolds with complex germs. To such a manifold we assign a canonical operator
that depends on creation and annihilation operators. This operator is by
definition the geometrical quantization for these isotropic manifolds with
complex germs. We prove that for secondary quantized equations this
quantization is the asymptotics for the Cauchy problem. Results of Berezin are
used thouroughly in the construction of the canonical operator and in proofs of
the theorems.


title:
Antiholomorphic representations for orthogonal and symplectic quantum
  groups

abstract:
  The coadjoint orbits for the series $B_l,\ C_l$ and $D_l$ are considered in
the case when the base point is a multiple of a fundamental weight. A
quantization of the big cell is suggested by means of introducing a
$\ast$-algebra generated by holomorphic coordinate functions. Starting from
this algebraic structure the irreducible representations of the deformed
universal enveloping algebra are derived as acting in the vector space of
polynomials in quantum coordinate functions.


title: L-infinity algebras and their cohomology

abstract:
  An associative algebra is nothing but an odd quadratic codifferential on the
tensor coalgebra of a vector space, and an A-infinity algebra is simply an
arbitrary odd codifferential. Hochschild cohomology classifies the deformations
of an associative algebra into an A-infinity algebra, and cyclic cohomology in
the presence of an invariant inner product classifies the deformations of the
associative algebra into an A-infinity algebra preserving the inner product.
Similarly, a graded Lie algebra is simply a special case of an odd
codifferential on the exterior coalgebra of a vector space, and an L-infinity
algebra is a more general codifferential. In this paper, ordinary and cyclic
cohomology of L-infinity algebras is defined, and it is shown that the
cohomology of a Lie algebra (with coefficients in the adjoint representation)
classifies the deformations of the Lie algebra into an L-infinity algebra.
Similarly, the cyclic cohomology of a Lie algebra with an invariant inner
product classifies the deformations of the Lie algebra into an L-infinity
algebra which preserve the invariant inner product. The exterior coalgebra of a
vector space is dual to the symmetric coalgebra of the parity reversion of the
space, while the tensor coalgebra of a vector space is dual to the tensor
coalgebra of its parity reversion. Using this duality, we introduce a modified
bracket in the space of coderivations of the tensor and exterior coalgebras
which makes it possible to treat the cohomology of an A-infinity or L-infinity
as a differential graded algebra in the same manner in which the Gerstenhaber
bracket is used to transform the Hochschild cochains of an associative algebra
into a differential graded algebra.


title: On Commutation Relations for Quons

abstract:
  The model of generalized quons is described in a purely algebraic way.
Commutation relations and corresponding consistency conditions for our
generalized quons system are studied in terms of quantum Weyl algebras. Fock
space representation and corresponding scalar product is also given.


title:
Sugawara Construction and Casimir Operators for Krichever-Novikov
  Algebras

abstract:
  We show how to obtain from highest weight representations of
Krichever-Novikov algebras of affine type (also called higher genus affine
Kac-Moody algebras) representations of centrally extended Krichever-Novikov
vector field algebras via the Sugawara construction. This generalizes classical
results where one obtains representations of the Virasoro algebra. Relations
between the weights of the corresponding representations are given and Casimir
operators are constructed. In an appendix the Sugawara construction for the
multi-point situation is done.


title: Quantum group covariant systems

abstract:
  The meaning of quantum group transformation properties is discussed in some
detail by comparing the (co)actions of the quantum group with those of the
corresponding Lie group, both of which have the same algebraic (matrix) form of
the transformation. Various algebras are considered which are covariant with
respect to the quantum (super) groups $SU_q(2),\; SU_q(1, 1),\; SU_q(1|1),\;
SU_q(n), \\ SU_q(m|n),\; OSp_q(1|2)$ as well as deformed Minkowski space-time
algebras.


title: The Kappa-Weyl group and its algebra

abstract:
  The $\kappa -$Poincare group and its algebra in an arbitrary basis are
constructed. The $\kappa -$de\-formation of the Weyl group and its algebra in
any dimensions and in the reference frame in which $g_{00}=0$ are discussed.


title:
Subalgebras with Converging Star Products in Deformation Quantization:
  An Algebraic Construction for $\complex \mbox{\LARGE P}^n$

abstract:
  Based on a closed formula for a star product of Wick type on $\CP^n$, which
has been discovered in an earlier article of the authors, we explicitly
construct a subalgebra of the formal star-algebra (with coefficients contained
in the uniformly dense subspace of representative functions with respect to the
canonical action of the unitary group) that consists of {\em converging} power
series in the formal parameter, thereby giving an elementary algebraic proof of
a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this
subalgebra the formal parameter can be substituted by a real number $\alpha$:
the resulting associative algebras are infinite-dimensional except for the case
$\alpha=1/K$, $K$ a positive integer, where they turn out to be isomorphic to
the finite-dimensional algebra of linear operators in the $K$th energy
eigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom.
Other examples like the $2n$-torus and the Poincar\'e disk are discussed.


title: Graded Contractions of Affine Kac-Moody Algebras

abstract:
  The method of graded contractions, based on the preservation of the
automorphisms of finite order, is applied to the affine Kac-Moody algebras and
their representations, to yield a new class of infinite dimensional Lie
algebras and representations. After the introduction of the horizontal and
vertical gradings, and the algorithm to find the horizontal toroidal gradings,
I discuss some general properties of the graded contractions, and compare them
with the In\"on\"u-Wigner contractions. The example of $\hat A_2$ is discussed
in detail.


title: Remarks on Geometric Quantization of R-matrix Type Poisson Brackets

abstract:
  We check The Vaisman condition of geometric quantization for R-matrix type
Poisson pencil on a coadjoint orbit of a compact Lie group. It is shown that
this condition isn't satisfied.


title:
The universal R-matrix for the Jordanian deformation of sl(2), and the
  contracted forms of so(4)

abstract:
  We introduce a universal R matrix for the Jordanian deformation of $\U{
\sl(2)}$. Using $\Uh{\so(4)}=\Uh{\sl(2)} \oplus {\rm U}_{-h}(\sl(2))$, we
obtain the universal R matrix for $\Uh{\so(4)}$. Applying the graded
contractions on the universal R matrix of $\Uh{\so(4)}$, we show that there
exist three distinct R matrices for all of the contracted algebras. It is shown
that $\Uh{\sl(2)}$, $\Uh{\so(4)}$, and all of these contracted algebras are
triangular.


title: A New Deformed Supersymmetric Oscillator

abstract:
  We construct and discuss the Fock-space representation for a deformed
oscillator with "peculiar" statistics. We show that corresponding algebra
represents deformed supersymmetric oscillator.


title:
Intertwining Operator Algebras, Genus-Zero Modular Functors and
  Genus-Zero Conformal Field Theories

abstract:
  We describe the construction of the genus-zero parts of conformal field
theories in the sense of G. Segal from representations of vertex operator
algebras satisfying certain conditions. The construction is divided into four
steps and each step gives a mathematical structure of independent interest.
These mathematical structures are intertwining operator algebras, genus-zero
modular functors, genus-zero holomorphic weakly conformal field theories, and
genus-zero conformal field theories.


title: The algebra of q-pseudodifferential symbols and the q-W_{KP}^n algebra

abstract:
  In this paper we continue with the program to explore the topography of the
space of W-type algebras. In the present case, the starting point is the work
of Khesin, Lyubashenko and Roger on the algebra of q-deformed
pseudodifferential symbols and their associated integrable hierarchies. The
analysis goes on by studying the associated hamiltonian structures for which
compact expressions are found. The fundamental Poisson brackets yield
q-deformations of W_{KP} and related W-type algebras which, in specific cases,
coincide with the ones constructed by Frenkel and Reshetikhin. The construction
underlies a continuous correspondence between the hamiltonian structures of the
Toda lattice and the KP hierarchies.


title: Decomposition of the adjoint representation of the small quantum $sl_2$

abstract:
  Given a finite type root datum and a primitive root of unity $q=\sqrt[l]{1}$,
G.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra
$\fu$ over the cyclotomic field ${\Bbb Q}(\sqrt[l]{1})$. In this note we study
the adjoint representation $\ad$ of $\fu$ in the simplest case of the root
datum $sl_2$. The semisimple part of this representation is of big importance
in the study of local systems of conformal blocks in WZW model for $\hat{sl}_2$
at level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple
part is closely related to the problem of integral representation of conformal
blocks (see [BFS]). We find all the indecomposable direct summands of $\ad$
with multiplicities. It appears that $\ad$ is isomorphic to a direct sum of
simple and projective modules. It can be lifted to a module over the (infinite
dimensional) quantum universal enveloping algebra with divided powers
$U_q(sl_2)$ which is also a direct sum of simples and projectives.


title: Kostka Polynomials and Energy Functions in Solvable Lattice Models

abstract:
  The relation between the charge of Lascoux-Schuzenberger and the energy
function in solvable lattice models is clarified. As an application,
A.N.Kirillov's conjecture on the expression of the branching coefficient of
${\widehat {sl_n}}/{sl_n}$ as a limit of Kostka polynomials is proved.


title: Some Comments on Bosonisation and Biproducts

abstract:
  We collect here some less well-known results and formulae about the
bosonisation construction which turns braided groups into quantum groups. We
clarify the relation with biproduct Hopf algebras (the constructions are not
the same), the response to twisting of braided groups and the abstract
characterisation via automorphisms of the forgetful functor for the category of
(co)modules of a braided group.


title:
Ruijsenaars' commuting difference operators as commuting transfer
  matrices

abstract:
  For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set
of difference operators acting on functions on the type A weight space.
According to the fundamental relation $RLL=LLR$, the trace of the L-operator
gives a commuting difference operators. We show that for the above mentioned
L-operator this approach gives Macdonald type operators with elliptic theta
function coefficient, actually equivalent to Ruijsenaars' operators. The
relationship between the difference L-operator and Krichever's Lax matrix as
well as other approaches for the elliptic commuting system is given. We also
study the invariant subspace for the system which is spanned by symmetric theta
functions.


title: Tensor operators in R-matrix approach

abstract:
  The definitions and some properties (e.g. the Wigner-Eckart theorem, the
fusion procedure) of covariant and contravariant q-tensor operators for
quasitriangular quantum Lie algebras are formulated in the R-matrix language.
The case of U_q(sl(n)) (in particular, for n=2) is discussed in more detail.


title:
Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and
  Unipotent Varieties

abstract:
  We introduce a new family of symmetric functions, which are $q$-analogues of
products of Schur functions defined in terms of ribbon tableaux. These
functions can be interpreted in terms of the Fock space representation of the
quantum affine algebra of type $A_{n-1}^{(1)}$ and are related to
Hall-Littlewood functions via the geometry of flag varieties. We present a
series of conjectures, and prove them in special cases. The essential step in
proving that these functions are actually symmetric consists in the calculation
of a basis of highest weight vectors of the $q$-Fock space using ribbon
tableaux.


title: Recursively minimally-deformed oscillators

abstract:
  A recursive deformation of the boson commutation relation is introduced. Each
step consists of a minimal deformation of a commutator
$[a,\ad]=f_k(\cdots;\no)$ into $[a,\ad]_{q_{k+1}}=f_k(\cdots;\no)$, where
$\cdots$ stands for the set of deformation parameters that $f_k$ depends on,
followed by a transformation into the commutator $[a,\ad]=f_{k+1}(\cdots,\,
q_{k+1};\no)$ to which the deformed commutator is equivalent within the Fock
space. Starting from the harmonic oscillator commutation relation $[a,\ad]=1$
we obtain the Arik-Coon and the Macfarlane-Biedenharn oscillators at the first
and second steps, respectively, followed by a sequence of multiparameter
generalizations. Several other types of deformed commutation relations related
to the treatment of integrable models and to parastatistics are also obtained.
The ``generic'' form consists of a linear combination of exponentials of the
number operator, and the various recursive families can be classified according
to the number of free linear parameters involved, that depends on the form of
the initial commutator.


title: Woven Braids and their Closures

abstract:
  A special class of braids, called woven, is introduced and it is shown that
every conjugation class of the braid group contains woven braids. In
consequence, links can be presented as plats or closures of woven braids.
Restricting on knots, we get the 'woven version' of the well-known theorem of
Markov, giving moves that are capable of producing all woven braids with
equivalent closures. As corollary we obtain that a link in which each component
is dyed with at least two colours can be projected onto a plane without
crossing strands of the same colour. Finally a table of minimal woven braids
for all nine crossing knots is appended.


title:
On Nonlinear Angular Momentum Theories, Their Representations and
  Associated Hopf Structures

abstract:
  Nonlinear $sl(2)$ algebras subtending generalized angular momentum theories
are studied in terms of undeformed generators and bases. We construct their
unitary irreducible representations in such a general context. The linear
$sl(2)$-case as well as its $q$-deformation are easily recovered as specific
examples. Two other physically interesting applications corresponding to the
so-called Higgs and quadratic algebras are also considered. We show that these
two nonlinear algebras can be equipped with a Hopf structure.


title:
An Invariant of Integral Homology 3-Spheres Which Is Universal For All
  Finite Type Invariants

abstract:
  In [LMO] a 3-manifold invariant $\Omega(M)$ is constructed using a
modification of the Kontsevich integral and the Kirby calculus. The invariant
$\Omega$ takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here
we show that for homology 3-spheres the invariant $\Omega$ is {\em universal}
for all finite type invariants, i.e. $\Omega_n$ is an invariant of order $3n$
which dominates all other invariants of the same order. This shows that the set
of finite type invariants of homology 3-spheres is equivalent to the Hopf
algebra of Feynman 3-valent graphs. Some corollaries are discussed. A theory of
groups of homology 3-spheres, similar to Gusarov's theory for knots, is
presented.


title: On representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$

abstract:
  We describe representation theory of the elliptic quantum group
$E_{\tau,\eta}(sl_2)$. It turns out that the representation theory is parallel
to the representation theory of the Yangian $Y(sl_2)$ and the quantum loop
group $ U_q(\widetilde { sl }_2)$.


title: Minimal model fusion rules from 2-groups

abstract:
  The fusion rules for the $(p,q)$-minimal model representations of the
Virasoro algebra are shown to come from the group $G = \boZ_2^{p+q-5}$ in the
following manner. There is a partition $G = P_1 \cup ...\cup P_N$ into disjoint
subsets and a bijection between $\{P_1,...,P_N\}$ and the sectors
$\{S_1,...,S_N\}$ of the $(p,q)$-minimal model such that the fusion rules $S_i
* S_j = \sum_k D(S_i,S_j,S_k) S_k$ correspond to $P_i * P_j = \sum_{k\in
T(i,j)} P_k$ where $T(i,j) = \{k|\exists a\in P_i,\exists b\in P_j, a+b\in
P_k\}$.


title: Representations of the deformed U(su(2)) and U(osp(1,2)) algebras

abstract:
  The polynomial deformations of the Witten extensions of the U(su(2)) and
U(osp(1,2)) algebras are three generator algebras with normal ordering,
admitting a two generator subalgebra. The modules and the representations of
these algebras are based on the construction of Verma modules, which are
quotient modules, generated by ideals of the original algebra. This
construction unifies a large number of the known algebras under the same
scheme. The finite dimensional representations show new features such as the
multiplicity of representations of the same dimensionality, or the existence of
finite dimensional representations only for some dimensions.


title:
Bicovariant Calculus on Twisted ISO(N), Quantum Poincare' Group and
  Quantum Minkowski Space

abstract:
  A bicovariant calculus on the twisted inhomogeneous multiparametric
$q$-groups of the $B_n,C_n,D_n$ type, and on the corresponding quantum planes,
is found by means of a projection from the bicovariant calculus on $B_{n+1}$,
$C_{n+1}$, $D_{n+1}$. In particular we obtain the bicovariant calculus on a
dilatation-free $q$-Poincar\'e group $ISO_q (3,1)$, and on the corresponding
quantum Minkowski space. The classical limit of the $B_n,C_n,D_n$ bicovariant
calculus is discussed in detail.


title: New Generalized Poisson Structures

abstract:
  New generalized Poisson structures are introduced by using suitable
skew-symmetric contravariant tensors of even order. The corresponding `Jacobi
identities' are provided by conditions on these tensors, which may be
understood as cocycle conditions. As an example, we provide the linear
generalized Poisson structures which can be constructed on the dual spaces of
simple Lie algebras.


title:
Central bialgebras in Braided Categories and Coquasitriangular
  Structures

abstract:
  Central bialgebras in a braided category $\C$ are algebras in the center of
the category of coalgebras in $\C$. On these bialgebras another product can be
defined, which plays the role of the opposite product. Hence, coquasitriangular
structures on central bialgebras can be defined. We prove some properties of
the antipode on coquasitriangular central Hopf algebras and give a
characterization of central bialgebras.


title:
Higher Order Terms in the Melvin-Morton Expansion of the Colored Jones
  Polynomial

abstract:
  We formulate a conjecture about the structure of `upper lines' in the
expansion of the colored Jones polynomial of a knot in powers of (q-1). The
Melvin-Morton conjecture states that the bottom line in this expansion is equal
to the inverse Alexander polynomial of the knot. We conjecture that the upper
lines are rational functions whose denominators are powers of the Alexander
polynomial. We prove this conjecture for torus knots and give experimental
evidence that it is also true for other types of knots.


title: Two-parameter deformation of the Poincar\'e algebra

abstract:
  We examine a two-parameter ($\hbar ,$ $\lambda $) deformation of the
Poincar\`e algebra which is covariant under the action of $SL_q(2,C).$ When
$\lambda \rightarrow 0$ it yields the Poincar\`e algebra, while in the
$\hbar\rightarrow 0$ limit we recover the classical quadratic algebra discussed
previously in \cite{ssy95}, \cite{sy95}. The analogues of the Pauli-Lubanski
vector $w$ and Casimirs $p^2$ and $w^2$ are found and a set of mutually
commuting operators is constructed.


title: Strings, paths, and standard tableaux

abstract:
  For the vacuum sectors of regime-III ABF models, we observe that two sets of
combinatorial objects: the strings which parametrize the row-to-row transfer
matrix eigenvectors, and the paths which parametrize the corner transfer matrix
eigenvectors, can both be expressed in terms of the same set of standard
tableaux. Furthermore, the momenta of the strings, the energies of the paths,
and the charges of the tableaux are such that there is a weight-preserving
bijection between the two sets of eigenvectors, wherein the tableaux play an
interpolating role. This bijection is so natural, that we conjecture that it
exists in general.


title: The Andrews-Gordon identities and $q$-multinomial coefficients

abstract:
  We prove polynomial boson-fermion identities for the generating function of
the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with
$f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of
the identities involves $q$-deformations of the coefficients of $x^a$ in the
expansion of $(1+x+\cdots+ x^k)^L$. A combinatorial interpretation for these
$q$-multinomial coefficients is given using Durfee dissection partitions. The
fermionic side of the polynomial identities arises as the partition function of
a one-dimensional lattice-gas of fermionic particles. In the limit
$L\to\infty$, our identities reproduce the analytic form of Gordon's
generalization of the Rogers--Ramanujan identities, as found by Andrews. Using
the $q \to 1/q$ duality, identities are obtained for branching functions
corresponding to cosets of type $({\rm A}^{(1)}_1)_k \times ({\rm
A}^{(1)}_1)_{\ell} / ({\rm A}^{(1)}_1)_{k+\ell}$ of fractional level $\ell$.


title:
On the absolute value of the SO(3)-invariant and other summands of the
  Turaev-Viro invariant

abstract:
  The Turaev-Viro invariant is defined as a certain state sum calculated on an
arbitrary simple spine of a 3-manifold. We specify each term of the sum as
0-term, 1-term or 2-term such that each sum of the terms having the same type
is an invariant too. The sum of the 0-terms is equal to the square of the
modulus of the so-called SO(3)-invariant. In the paper we express the sum of
the 0-terms and 2-terms and the sum of the 1-terms via the Turaev-Viro
invariants. Tables of values of the invariants are enclosed. The values are
presented as polynomials on primitive roots of unity with integer coefficients.


title: Remarks on Quantum Integration

abstract:
  We give a general integration prescription for finite dimensional braided
Hopf algebras, deriving the N-dimensional quantum superplane integral as an
example. The transformation properties of the integral on the quantum plane are
found. We also discuss integration on quantum group modules that lack a Hopf
structure.


title:
On p-Adic Convergence of Perturbative Invariants of Some Rational
  Homology Spheres

abstract:
  R.~Lawrence has conjectured that for rational homology spheres, the series of
Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev
invariant. We prove this conjecture for Seifert rational homology spheres. We
also derive it for manifolds constructed by a surgery on a knot in S^3. Our
derivation is based on a conjecture about the colored Jones polynomial that we
have formulated in our previous paper. We also present numerical examples of
p-adic convergence for some simple manifolds.


title: Berezin-Toeplitz Quantization of compact Kaehler manifolds

abstract:
  Invited lecture at the XIV-th workshop on geometric methods in physics,
Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed
obtained by the author together with Martin Bordemann and Eckhard Meinrenken on
the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global
Toeplitz operators, approximation results for the quantum operators are shown.
From them it follows that the quantum operators have the correct classical
limit. A star product deformation of the Poisson algebra is constructed.

categories: q-alg alg-geom dg-ga funct-an math-ph math.AG math.DG math.FA math.MP math.QA


title:
Highest weight modules of W_{1+infty}, Darboux transformations and the
  bispectral problem

abstract:
  We announce a systematic way for constructing bispectral algebras of
commuting differential operators of any rank N. It enables us to obtain all
previously known classes and examples of bispectral operators. Moreover, we
give a representation-theoretic explanation of the results including those of
Duistermaat and Gr\"unbaum. The manifold of bispectral operators of any order
is preserved by an hierarchy of symmetries. We point out that our methods
provide a completely algorithmic procedure for obtaining bispectral algebras.
We conjecture that the class built in the present paper exhausts all bispectral
scalar operators.
  The proofs and details appeared in our preprints hep-th/9510211,
q-alg/9602010, q-alg/9602011, q-alg/9602012.


title: Infinity Algebras and the Homology of Graph Complexes

abstract:
  An A-infinity algebra is a generalization of a associative algebra, and an
L-infinity algebra is a generalization of a Lie algebra. In this paper, we show
that an L-infinity algebra with an invariant inner product determines a cycle
in the homology of the complex of metric ordinary graphs. Since the cyclic
cohomology of a Lie algebra with an invariant inner product determines
infinitesimal deformations of the Lie algebra into an L-infinity algebra with
an invariant inner product, this construction shows that a cyclic cocycle of a
Lie algebra determines a cycle in the homology of the graph complex. In this
paper a simple proof of the corresponding result for A-infinity algebras, which
was proved in a different manner in an earlier paper, is given.


title: Finite type link invariants and the non-invertibility of links

abstract:
  We show that a variation of Milnor's $\bar\mu$-invariants, the so-called
Campbell-Hausdorff invariants introduced recently by Stefan Papadima, are of
finite type with respect to {\it marked singular links}. These link invariants
are stronger than quantum invariants in the sense that they detect easily the
non-invertibility of links with more than one components. It is still open
whether some effectively computable knot invariants, e.g. finite type knot
invariants (Vassiliev invariants), could detect the non-invertibility of knots.


title: Left-Covariant Differential Calculi on SL_q(2) and SL_q(3)

abstract:
  We study $N^2-1$ dimensional left-covariant differential calculi on the
quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras
annihilate the quantum trace. In this way we obtain one distinguished calculus
on $SL_q(2)$ (which corresponds to Woronowicz' 3D-calculus on $SU_q(2)$) and
two distinguished calculi on $SL_q(3)$ such that the higher order calculi give
the ordinary differential calculus on $SL(2)$ and $SL(3)$, respectively, in the
limit $q\rightarrow 1$. Two new differential calculi on $SL_q(3)$ are
introduced and developed in detail.


title:
Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The
  Surgical Family

abstract:
  We study the relations between the invariants $\tau_{RT}$, $\tau_{HKR}$, and
  $\tau_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko,
  respectively. In particular, we discuss explicitly how $\tau_L$ specializes
to $\tau_{RT}$ for semisimple categories and to $\tau_{HKR}$ for Tannakian
categories. We give arguments for that $\tau_L$ is the most general invariant
that stems from an extended TQFT. We introduce a canonical, central element,
{\sf Q}, for a quasi-triangular Hopf algebra, $\A$, that allows us to apply the
Hennings algorithm directly, in order to compute $\tau_{RT}$, which is
originally obtained from the semisimple trace-subquotient of $\A-mod$.
Moreover, we generalize Hennings' rules to the context of cobordisms, in order
to obtain a TQFT for connected surfaces compatible with $\tau_{HKR}\,$. As an
application we show that, for lens spaces and $\A=U_q(sl_2)\,$, the ratio of
$\tau_{HKR}$ and $\tau_{RT}$ is the order of the first homology group. In the
course of this paper we also outline the topology and the algebra that enter
invariance proofs, which contain no reference to 2-handle slides, but to other
moves that are local. Finally, we give a list of open questions regarding
cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their
relations among each other, and their relations to the surgical invariants from
above.


title:
Quantum groups in higher genus and Drinfeld's new realizations method
  ($sl_{2}$ case)

abstract:
  We define double (central and cocentral) extensions of Manin pairs introduced
by Drinfeld, attached to curves and meromorphic differentials. We define
``infinite twistings'' of these pairs and quantize them in the $sl_{2}$ case,
adapting Drinfeld's ``new realizations'' technique. We study finite dimensional
representations of these algebras in level $0$, and some elliptic examples.


title: Differential Calculi and Linear Connections

abstract:
  A method is proposed for defining an arbitrary number of differential calculi
over a given noncommutative associative algebra. As an example the generalized
quantum plane is studied. It is found that there is a strong correlation, but
not a one-to-one correspondence, between the module structure of the 1-forms
and the metric torsion-free connections on it. In the commutative limit the
connection remains as a shadow of the algebraic structure of the 1-forms.


title: Double quantum groups and Iwasawa decomposition

abstract:
  The double quantum groups are the Hopf algebras underlying the complex
quantum groups of which the simplest example is the quantum Lorentz group. They
are non- standard quantizations of the double group $G \times G$. We construct
a corresponding quantized universal enveloping algebra (QUE) and prove that the
pairing between a quantum double group and its QUE is nondegenerate. We analyze
the representation theory of these double quantum groups, give a detailed
version of the Iwasawa decomposition proved by Podles and Woronowicz for the
quantum Lorentz group, and show that they are noetherian algebras. Finally we
outline a construction of more general non-standard quantum groups using
quantum double groups and their generalizations.


title: The hyperbolic volume of knots from quantum dilogarithm

abstract:
  The invariant of a link in three-sphere, associated with the cyclic quantum
dilogarithm, depends on a natural number $N$. By the analysis of particular
examples it is argued that for a hyperbolic knot (link) the absolute value of
this invariant grows exponentially at large $N$, the hyperbolic volume of the
knot (link) complement being the growth rate.


title: Semiinfinite cohomology of quantum groups

abstract:
  In this paper we develop a new homology theory of associative algebras called
semiinfinite cohomology in a derived category setting. We show that in the case
of small quantum groups the zeroth semiinfinite cohomology of the trivial
module is closely related to the conformal blocks' spaces. We provide a
description of the semiinfinite cohomology spaces of the trivial module over a
small quantum group in terms of distributions on the nilpotent cone of the
corresponding semisimple Lie algebra.


title: Yangians and Capelli identities

abstract:
  We study the image of the universal $R$-matrix for the Yangian $Y(gl_N)$ with
respect to the evaluation homomorphism of $Y(gl_N)$ to the enveloping algebra
$U(gl_N)$. We use the fusion procedure as defined by I. Cherednik. As a
corollary we get a generalization of the classical Capelli identity.


title: Algebraic Bethe Ansatz for XYZ Gaudin model

abstract:
  The eigenvectors of the Hamiltionians of the XYZ Gaudin model are constructed
by means of the algebraic Bethe Ansatz. The construction is based on the
quasi-classical limit of the corresponding results for the inhomogeneous higher
spin eight vertex model.


title: An Algorithmic Approach to Classify Knots

abstract:
  I briefly discuss a method of obtaining distinct classes of topologically
equivalent knots by developing appropriate computer programs.


title: Braided Geometry of the Conformal Algebra

abstract:
  We show that the action of the special conformal transformations of the usual
(undeformed) conformal group is the $q\to 1$ scaling limit of the braided
adjoint action or $R$-commutator of $q$-Minkowski space on itself. We also
describe the $q$-deformed conformal algebra in $R$-matrix form and its
quasi-$*$ structure.


title: Poisson structures on the Poincare group

abstract:
  An introduction to inhomogeneous Poisson groups is given. Poisson
inhomogeneous $O(p,q)$ are shown to be coboundary, the generalized classical
Yang-Baxter equation having only one-dimensional right hand side. Normal forms
of the classical $r$-matrices for the Poincar\'{e} group (inhomogeneous
$O(1,3)$) are calculated.


title: A characterization of coboundary Poisson Lie groups and Hopf algebras

abstract:
  We show that a Poisson Lie group $(G,\pi)$ is coboundary if and only if the
natural action of $G\times G$ on $M=G$ is a Poisson action for an appropriate
Poisson structure on $M$ (the structure turns out to be the well known $\pi
_+$). We analyze the same condition in the context of Hopf algebras. Quantum
analogue of the $\pi_+$ structure on SU(N) is described in terms of generators
and relations as an example.


title:
Contraction of Algebraical Structures and Different Couplings of
  Cayley-Klein and Hopf Structures

abstract:
  Contractions (and graded contractions) of Lie algebra, Lie bialgebra and Hopf
algebra are discussed. It is noticed the fundamental role of E.In{\"o}n{\"u}
and E.P.Wigner idea of degenerate transformations. A constructive algorithm for
description of contractions of quantum Cayley-Klein algebras $ so_{z}(n+1; {\bf
j}) $ with different choice of the set of primitive operators is suggested. For
nonsemisimple quantum algebras it gives nonisomorphic Hopf algebras. From
physical point of view this algorithm gives the different physical
interpretations of primitive operators for mathematically the same
nonsemisimple quantum algebra. The case of $ so_{z}(3; {\bf j}) $ is regarded
in detail.


abstract:
  The bicovariant differential calculus on the three-dimensional
Kappa-Poincar'e group and the corresponding Lie-algebra structure are
described. The equivalence of this Lie-algebra structure and the
three-dimensional $\kappa$-Poincar\'e algebra is proved.


title: Deformation map for generalized Kappa-Poincar'e and Kappa-Weyl algebras

abstract:
  A nonlinear transformation in the momentum space is constructed which
converts the deformed action of Lorentz and Weyl generators on momenta into the
standard one.


title: A note on geometry of Kappa-Minkowski space

abstract:
  The infinitesimal action of Kappa-Poincar'e group on Kappa-Minkowski space is
computed both for generators of Kappa-Poincar'e algebra and those of Woronowicz
generalized Lie algebra. The notion of invariant operators is introduced and
generalized Klein-Gordon equation is written out.


title: Differential calculi on quantum Minkowski space

abstract:
  The differential calculus on n-dimensional quantum Minkowski space covariant
with respect to left action of Kappa-Poincar'e group is constructed and its
uniqueness is shown.


title: Differential calculus on the quantum Heisenberg group

abstract:
  The differential calculus on the quantum Heisenberg group is conlinebreak
structed. The duality between quantum Heisenberg group and algebra is proved.


title: Homotopy Gerstenhaber algebras and topological field theory

abstract:
  We prove that the BRST complex of a topological conformal field theory is a
homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We
also suggest a refinement of the original conjecture for topological vertex
operator algebras. We illustrate the usefulness of our main tools, operads and
"string vertices" by obtaining new results on Vassiliev invariants of knots and
double loop spaces.


abstract:
  We define B\"acklund--Darboux transformations in Sato's Grassmannian. They
can be regarded as Darboux transformations on maximal algebras of commuting
ordinary differential operators. We describe the action of these
transformations on related objects: wave functions, tau-functions and spectral
algebras.
  This paper is the second of a series of papers (hep-th/9510211,
q-alg/9602011, q-alg/9602012) on the bispectral problem.


title: Bispectral algebras of commuting ordinary differential operators

abstract:
  We develop a systematic way for constructing bispectral algebras of commuting
ordinary differential operators of any rank $N$. It combines and unifies the
ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely
algorithmic and enables us to obtain all previously known classes or individual
examples of bispectral operators. The method also provides new broad families
of bispectral algebras which may help to penetrate deeper into the problem.


title:
Highest weight modules over W_{1+infty} algebra and the bispectral
  problem

abstract:
  The present paper establishes a connection between the Lie algebra
W_{1+infty} and the bispectral problem. We show that the manifolds of
bispectral operators obtained by Darboux transformations on powers of Bessel
operators are in one to one correspondence with the manifolds of tau-functions
lying in the W_{1+infty}-modules M_beta introduced in our previous paper
hep-th/9510211. An immediate corollary is that they are preserved by
hierarchies of symmetries generated by subalgebras of W_{1+infty}.
  This paper is the last of a series of papers (hep-th/9510211, q-alg/9602010,
q-alg/9602011) on the bispectral problem.


title: Semiinfinite cohomology of associative algebras and bar duality

abstract:
  We describe semiinfinite cohomology of associative algebras in terms of
Koszul (or bar) duality. Consider an associative algebra $A$ and two its
subalgebras $B$ and $N$ such that $A=B\otimes N$ as a vector space. We prove
that the endomorphism algebra of the semiregular $A$-module appears naturally
in semiinfinite cohomology theory as a ``two times Koszul dual'' to the algebra
$A$. We compare semiinfinite cohomology of universal enveloping algebras with
the well-known Lie algebra semiinfinite cohomology. A new description of the
critical 2-cocycle is provided. As a consequence we obtain another proof of the
fact that additive categories generated by Verma modules over affine Lie
algebras on dual levels are antiequivalent.


title: The universal Vassiliev invariant for the Lie superalgebra gl(1|1)

abstract:
  We compute the universal weight system for Vassiliev invariants coming from
the Lie superalgebra gl(1|1) applying the construction of \cite{YB}. This
weight system is a function from the space of chord diagrams to the center $Z$
of the universal enveloping algebra of gl(1|1), and we find a combinatorial
expression for it in terms of the standard generators of $Z$. The resulting
knot invariants generalize the Alexander-Conway polynomial.


title: Cabled Wilson Loops in BF Theories

abstract:
  A generating function for cabled Wilson loops in three-dimensional BF
theories is defined, and a careful study of its behavior for vanishing
cosmological constant is performed. This allows an exhaustive description of
the unframed knot invariants coming from the pure BF theory based on SU(2), and
in particular, it proves a conjecture relating them to the Alexander-Conway
polynomial.


title: Contractions, Hopf algebra extensions and cov. differential calculus

abstract:
  We re-examine all the contractions related with the ${\cal U}_q(su(2))$
deformed algebra and study the consequences that the contraction process has
for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$
as an example that, as in the undeformed case, the contraction may generate
Hopf algebra cohomology. We shall show that most of the different Hopf algebra
deformations obtained have a bicrossproduct or a cocycle bicrossproduct
structure, for which we shall also give their dual `group' versions. The
bicovariant differential calculi on the deformed spaces associated with the
contracted algebras and the requirements for their existence are examined as
well.


title:
Comparing finite type invariants of knots and integral homology
  3-spheres

abstract:
  Using elementary counting methods of weight systems for finite type
invariants of knots and integral homology 3-spheres, in the spirit of [B-NG],
we answer positively three questions raised in [Ga]. In particular, we exhibit
a one-to-one map from the space of finite type invariants of integral homology
3-spheres to the space of finite type invariants of knots in $S^3$.


title: Demazure modules and vertex models: the affine sl(2) case

abstract:
  We characterize, in the case of affine sl(2), the crystal base of the
Demazure module E_w(\La) in terms of extended Young diagrams or paths for any
dominant integral weight \La and Weyl group element w. Its character is
evaluated via two expressions, 'bosonic' and 'fermionic'.


title: Interesting Relations in Fock Space

abstract:
  Certain non-linear relations between the generators of the (q-deformed)
Heisenberg algebra are found. Some of these relations are invariant under
quantization and $q$-deformation.


title:
The dual $(p,q)$-Alexander-Conway Hopf algebras and the associated
  universal ${\cal T}$-matrix

abstract:
  The dually conjugate Hopf algebras $Fun_{p,q}(R)$ and $U_{p,q}(R)$ associated
with the two-parametric $(p,q)$-Alexander-Conway solution $(R)$ of the
Yang-Baxter equation are studied. Using the Hopf duality construction, the full
Hopf structure of the quasitriangular enveloping algebra $U_{p,q}(R)$ is
extracted. The universal ${\cal T}$-matrix for $Fun_{p,q}(R)$ is derived. While
expressing an arbitrary group element of the quantum group characterized by the
noncommuting parameters in a representation independent way, the ${\cal
T}$-matrix generalizes the familiar exponential relation between a Lie group
and its Lie algebra. The universal ${\cal R}$-matrix and the FRT matrix
generators, $L^{(\pm )}$, for $U_{p,q}(R)$ are derived from the ${\cal
T}$-matrix.


title: Spinor representations of $U_q(\hat{\frak o}(N))$

abstract:
  This is an extension of quantum spinor construction of $U_q(\hat {\frak
gl}(n))$. We define quantum affine Clifford algebras based on the tensor
category and the solutions of q-KZ equations, and construct quantum spinor
representations of $U_q(\hat{\frak o}(N))$.


title: Coalgebra Gauge Theory

abstract:
  We develop a generalised gauge theory in which the role of gauge group is
played by a coalgebra and the role of principal bundle by an algebra. The
theory provides a unifying point of view which includes quantum group gauge
theory, embeddable quantum homogeneous spaces and braided group gauge theory,
the latter being introduced now by these means. Examples include ones in which
the gauge groups are the braided line and the quantum plane.


title:
Separation of variables for A2 Ruijsenaars model and new integral
  representation for A2 Macdonald polynomials

abstract:
  Using the Baker-Akhiezer function technique we construct a separation of
variables for the classical trigonometric 3-particle Ruijsenaars model
(relativistic generalization of Calogero-Moser-Sutherland model). In the
quantum case, an integral operator M is constructed from the Askey-Wilson
contour integral. The operator M transforms the eigenfunctions of the commuting
Hamiltonians (Macdonald polynomials for the root sytem A2) into the factorized
form S(y1)S(y2) where S(y) is a Laurent polynomial of one variable expressed in
terms of the 3phi2(y) basic hypergeometric series. The inversion of M produces
a new integral representation for the A2 Macdonald polynomials. We also present
some results and conjectures for general n-particle case.

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


title: Massey products and deformations

abstract:
  The classical deformation theory of Lie algebras involves different kinds of
Massey products of cohomology classes. Even the condition of extendibility of
an infinitesimal deformation to a formal one-parameter deformation of a Lie
algebra involves Massey powers of two dimensional cohomology classes which are
not powers in the usual definition of Massey products in the cohomology of a
differential graded Lie algebra. In the case of deformations with other local
bases, one deals with other, more specific Massey products. In the present work
a construction of generalized Massey products is given, depending on an
arbitrary graded commutative, associative algebra. In terms of these products,
the above condition of extendibility is generalized to deformations with
arbitrary local bases. Dually, a construction of generalized Massey products on
the cohomology of a differential graded commutative associative algebra depends
on a nilpotent graded Lie algebra. For example, the classical Massey products
correspond to the Lie algebra of strictly upper triangular matrices, while the
matric Massey products correspond to the Lie algebra of block strictly upper
triangular matrices.


title: Canonical bases of q-deformed Fock spaces

abstract:
  We define a canonical basis of the $q$-deformed Fock space representation of
the affine Lie algebra $\glchap_n$. We conjecture that the entries of the
transition matrix between this basis and the natural basis of the Fock space
are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$
specialized to a $n$th root of unity.


title: Tensor product of Vertex operator algebras

abstract:
  Let $V_1 \otimes V_2$ be a tensor product of VOAs. Using Zhu theory we
discuss the theory of representations of V (associative algebra, modules and
fusion rules). We prove that this theory is more or less the same as
representation theory of tensor product of the associative algebras.


title: Young Basis, Wick Formula, and Higher Capelli Identities

abstract:
  We prove Capelli type identities which involve the whole universal enveloping
algebra $U(gl(n))$ and matrix elements of irreducible representations of the
symmetric group. These identities generalize higher Capelli identities for the
center of $U(gl(n))$ introduced in the author's paper \cite{Ok}. The main role
in the proof play the Jucis-Murphy elements.


title: Quantum Immanants and Higher Capelli Identities

abstract:
  We consider remarkable central elements of the universal enveloping algebra
of the general linear algebra which we call quantum immanants. We express them
in terms of generators $E_{ij}$ and as differential operators on the space of
matrices. These expressions are a direct generalization of the classical
Capelli identities. They result in many nontrivial properties of quantum
immanants.


title:
Lie bialgebra quantizations of the oscillator algebra and their
  universal $R$--matrices

abstract:
  All coboundary Lie bialgebras and their corresponding Poisson--Lie structures
are constructed for the oscillator algebra generated by $\{\aa,\ap,\am,\bb\}$.
Quantum oscillator algebras are derived from these bialgebras by using the
Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both
algebra and group levels are obtained, including their universal $R$--matrices.


title: Intertwining Operators for the Central Extension of the Yangian Double

abstract:
  We continue the investigation of the central extended Yangian double [S.
Khoroshkin, q-alg/9602031]. In this paper we study the intertwining operators
for certain infinite dimensional representations of $\Yd$, which are deformed
analogs of the highest weight representations of the affine algebra $\aff$ at
level 1. We give bosonized expressions for intertwining operators, verify that
they generate an algebra isomorphic to Zamolodchikov--Faddeev algebra for the
$SU(2)$-invariant Thirring model. We compose from them $L$-operators by Miki's
prescription and verify that they coincide with $L$-operators constructed from
universal ${\cal R}$-matrix.


title: Central Extension of the Yangian Double

abstract:
  Central extension $\DYg$ of the Double of the Yangian is defined for a simple
Lie algebra ${\bf g}$ with complete proof for ${\bf g} =sl_2$. Basic
representations and intertwining operators are constructed for $\DY2$.


title: Deformed and extended Galilei group Hopf algebras

abstract:
  The $\hat\kappa$-deformed extended Galilei Hopf group algebra, ${\rm
Fun}_{\hat\kappa}(\tilde G_{(m)})$, is introduced. It provides an example of a
cocycle bicrossproduct structure, and is shown to be the contraction limit of a
pseudoextension of the $\kappa$-Poincar\'e group algebra. The possibility of
obtaining another deformed extended Galilei groups is discussed, including one
obtained from a non-standard Poincar\'e deformation.


title: The multiple gamma function and its q-analogue

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


title: Exact Deformations of Quantum Groups; Applications to the Affine case

abstract:
  This paper continues our investigation of a class of generalized quantum
groups. The "standard" R-matrix was shown to be the unique solution of a very
simple, linear recursion relation and the classical limit was obtained in the
case of quantized Kac-Moody algebras of finite type. Here the standard R-matrix
for generalized quantum groups is first examined in the case of quantized
affine Kac-Moody algebras. The classical limit yields the standard affine
r-matrices of Belavin and Drinfeld. Then, turning to the general case, we study
the exact deformations of the standard R-matrix and the associated Hopf
algebras. They are described as a generalized twist, $ R_\epsilon =
(F^t)^{-1}RF$, where $R$ is the standard R-matrix and $F$ (a power series in
the deformation parameter $\epsilon$) is the solution of a linear recursion
relation of the same type as that which determines $R$. Specializing again, to
the case of quantized, affine Kac-Moody algebras, and taking the classical
limit of these esoteric quantum groups, one re-discovers the esoteric affine
r-matrices of Belavin and Drinfeld, including the elliptic ones. The formulas
obtained here are easier to use than the original ones, and the structure of
the space of classical r-matrices (for simple Lie algebras) is more tranparent.
In addition, the r-matrices obtained here are more general in that they are
defined on the central extension of the loop groups.


title: Metrics and Pairs of Left and Right Connections on Bimodules

abstract:
  Properties of metrics and pairs consisting of left and right connections are
studied on the bimodules of differential 1-forms. Those bimodules are obtained
from the derivation based calculus of an algebra of matrix valued functions,
and an $SL\sb q(2,\IC)$-covariant calculus of the quantum plane plane at a
generic $q$ and the cubic root of unity. It is shown that, in the
aforementioned examples, giving up the middle-linearity of metrics
significantly enlarges the space of metrics. A~metric compatibility condition
for the pairs of left and right connections is defined. Also, a compatibility
condition between a left and right connection is discussed. Consequences
entailed by reducing to the centre of a bimodule the domain of those conditions
are investigated in detail. Alternative ways of relating left and right
connections are considered.


title: The centre of the graph and moduli algebras at roots of 1

abstract:
  The structure of the centres ${\cal Z}(\Lg)$ and ${\cal Z}(\Mg)$ of the graph
algebra ${\cal L}_g(sl_2)$ and the moduli algebra ${\cal M}_g(sl_2)$ is studied
at roots of 1. It it shown that ${\cal Z}(\Lg)$ can be endowed with the
structure of the Poisson graph algebra. The elements of $Spec({\cal Z}(\Mg))$
are shown to satisfy the defining relation for the holonomies of a flat
connection along the cycles of a Riemann surface. The irreducible
representations of the graph algebra are constructed.


title: Quasi-Classical Lie-Super Algebra and Lie-Super Triple Systems

abstract:
  Notions of quasi-classical Lie-super algebra as well as Lie-super triple
systems have been given and studied with some examples. Its application to
Yang-Baxter equation has also been given.


title: A note on Connections and Bimodules

abstract:
  We discuss three problems related to connections on bimodules. These are left
and right Leibniz rule for connections, left and right linearity of their
curvatures and extension of connections to tensor products of modules.


title:
Topological quantum field theory with corners based on the Kauffman
  bracket

abstract:
  The paper contains the construction of a topological quantum field theory
with corners that underlies the smooth topological quantum field theory of
Lickorish. Among other things, a contraction formula for diagrams is proved,
the presence of the skein relation at the level of manifolds with boundary is
shown and an explanation for the surgery formula for invariants of closed
manifolds is given.


title: Beyond the `Pentagon Identity'

abstract:
  An algebraical background of the Lattice Conformal Field Theory is refined
with the help of a novel $q$-exponential identity.


title:
Non Abelian Sugawara Construction and the q-deformed N=2 Superconformal
  Algebra

abstract:
  The construction of a q-deformed N=2 superconformal algebra is proposed in
terms of level 1 currents of ${\cal{U}}_{q} ({\widehat{su}}(2))$ quantum affine
Lie algebra and a single real Fermi field. In particular, it suggests the
expression for the q-deformed Energy-Momentum tensor in the Sugawara form. Its
constituents generate two isomorphic quadratic algebraic structures. The
generalization to ${\cal{U}}_{q} ({\widehat{su}}(N+1))$ is also proposed.


title: Generalized weight functions and the Macdonald polynomials

abstract:
  A weight function which $q$-generalizes the ground state wave function of the
multi-component Calogero-Sutherland quantum many body system is introduced.
Conjectures, and some proofs in special cases, are given for a constant term
identity involving this function. A Gram-Schmidt procedure with respect to the
inner product associated with the weight function is used to define orthogonal
polynomials in one of the components, which are conjectured to be the Macdonald
polynomials $P_\kappa(w_1,\dots,w_{N_0};qt^p,t)$, and a proof is given in a
special case. Conjectures are also given for an adjoint property of the
Macdonald operator with respect to the inner product associated with the weight
function, and the normalization of the Macdonald polynomial with respect to the
same inner product.


title: Equivalence of $q$-bosons using the exponential phase operator

abstract:
  Various forms of the $q$-boson are explained and their hidden symmetry
revealed by transformations using the exponential phase operator. Both the
one-component and the multicomponent $q$-bosons are discussed. As a byproduct,
we obtain a new boson algebra having a shifted vacuum structure and define a
global operatal $U(1)$ gauge transformation.


title: A Remark on the Higher Capelli Identities

abstract: A simple proof of the higher Capelli identities is given.


title: Kappa-contraction from $SU_q(2)$ to $E_{\kappa}(2)$

abstract:
  We present contraction prescription of the quantum groups: from $SU_q(2)$ to
$E_{\kappa}(2)$. Our strategy is different then one chosen in ref. [P. Zaugg,
J. Phys. A {\bf 28} (1995) 2589]. We provide explicite prescription for
contraction of $a, b, c$ and $d$ generators of $SL_q(2)$ and arrive at $^*$
Hopf algebra $E_{\kappa}(2)$.


title: Quantum E(2) groups and Lie bialgebra structures

abstract:
  Lie bialgebra structures on $e(2)$ are classified. For two Lie bialgebra
structures which are not coboundaries (i.e. which are not determined by a
classical $r$-matrix) we solve the cocycle condition, find the Lie-Poisson
brackets and obtain quantum group relations. There is one to one correspondence
between Lie bialgebra structures on $e(2)$ and possible quantum deformations of
$U(e(2))$ and $E(2)$.


title:
Vassiliev knot invariants and Chern-Simons perturbation theory to all
  orders

abstract:
  At any order, the perturbative expansion of the expectation values of Wilson
lines in Chern-Simons theory gives certain integral expressions. We show that
they all lead to knot invariants. Moreover these are finite type invariants
whose order coincides with the order in the perturbative expansion. Together
they combine to give a universal Vassiliev invariant.


title: Alexander-Conway limits of many Vassiliev weight systems

abstract:
  Previous work has shown that certain leading orders of arbitrary Vassiliev
invariants are generically in the algebra of the coefficients of the
Alexander-Conway polynomial \cite{KSA}. Here we illustrate this for a large
class of examples, exposing the simple logic behind several existing results in
the literature \cite{FKV,BNG}. This approach facilitates an extension to a
large class of Lie (super)algebras.


title: Hidden Symmetry of the Differential Calculus on the Quantum Matrix Space

abstract:
  A standard bicovariant differential calculus on a quantum matrix space ${\tt
Mat}(m,n)_q$ is considered. The principal result of this work is in observing
that the $U_q\frak{s}(\frak{gl}_m\times \frak{gl}_n))_q$ is in fact a
$U_q\frak{sl}(m+n)$-module differential algebra.


title:
Finite type 3-manifold invariants and the structure of the Torelli group
  I

abstract:
  Using the recently developed theory of finite type invariants of integral
homology 3-spheres we study the structure of the Torelli group of a closed
surface. Explicitly, we construct (a) natural cocycles of the Torelli group
(with coefficients in a space of trivalent graphs) and cohomology classes of
the abelianized Torelli group; (b) group homomorphisms that detect (rationally)
the nontriviality of the lower central series of the Torelli group. Our results
are motivated by the appearance of trivalent graphs in topology and in
representation theory and the dual role played by the Casson invariant in the
theory of finite type invariants of integral homology 3-spheres and in Morita's
study of the structure of the Torelli group Our results generalize those of S.
Morita {Mo1,Mo2} and complement the recent calculation, due to R. Hain, of the
I-adic completion of the rational group ring of the Torelli group. We also give
analogous results for two other subgroups of the mapping class group.


title: Multivariable Askey-Wilson Polynomials and Quantum Complex Grassmannians

abstract:
  We present a one-parameter family of constant solutions of the reflection
equation and define a family of quantum complex Grassmannians endowed with a
transitive action of the quantum unitary group. 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 multivariable Askey-Wilson polynomials
containing two continuous and two discrete parameters.


title:
On the defining relations of the affine Lie superalgebras and their
  quantized universal enveloping superalgebras

abstract:
  In this paper, we give defining relations of the affine Lie superalgebras an
and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine
quantized universal enveloping algebras. As a result, we can exactly define the
affine quantized universal enveloping superalgebras with generators and
relations. Moreover we give a Drinfeld's realizations of $U_h({\hat
{sl}}(m|n)^{(1)})$.


title: Crossed Products by a Coalgebra

abstract:
  We introduce the notion of a crossed product of an algebra by a coalgebra
$C$, which generalises the notion of a crossed product by a bialgebra
well-studied in the theory of Hopf algebras. The result of such a crossed
product is an algebra which is also a right $C$-comodule. We find the necessary
and sufficient conditions for two coalgebra crossed products be equivalent. We
show that the two-dimensional quantum Euclidean group is a coalgebra crossed
product. The paper is completed with an appendix describing the dualisation of
construction of coalgebra crossed products.


title: On the Connectivity of Cobordisms and Half-Projective TQFT's

abstract:
  We consider a generalization of the axioms of a TQFT, so called
half-projective TQFT's, with an anomaly, $x^{\mu}$, in the composition law.
$\mu$ is a coboundary on the cobordism categories with non-negative, integer
values. The element $x$ of the ring over which the TQFT is defined does not
have to be invertible. In particular, it may be 0. This modification makes it
possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to
non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be
trivial all together). Under a few natural assumptions the notion of a
half-projective TQFT is shown to be the only possible generalization. Based on
separate work with Lyubashenko on connected TQFT's, we construct a large class
of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times
S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf
algebras. Several toplogical tools that are relevant for vanishing properties
of such TQFT's are developed. They are concerned with connectivity properties
of cobordisms, as for example maximal non-separating surfaces. We introduce in
particular the notions of ``interior'' homotopy and homology groups, and of
coordinate graphs, which are functions on cobordisms with values in the
morphisms of a graph category. For applications we will prove that
half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior
homology, and we argue that the order of divergence of the TQFT on a cobordism
in the ``classical limit'' can be estimated by the rank of its maximal free
interior group.


title: Diagrammatics of Braided Group Gauge Theory

abstract:
  We develop a gauge theory or theory of bundles and connections on them at the
level of braids and tangles. Extending recent algebraic work, we provide now a
fully diagrammatic treatment of principal bundles, a theory of global gauge
transformations, associated braided fiber bundles and covariant derivatives on
them. We describe the local structure for a concrete $Z_3$-graded or `anyonic'
realization of the theory.


title: Dual Structures in Non-Commutative Differential Algebras

abstract:
  The non-commutative algebraic analog of the moduli of vector and covector
fields is built. The structure of moduli of derivations of non-commutative
algebras are studied. The canonical coupling is introduced and the conditions
for appropriate moduli to be reflexive are obtained.


title: On the D-module and formal-variable approaches to vertex algebras

abstract:
  In a program to formulate and develop two-dimensional conformal field theory
in the framework of algebraic geometry, Beilinson and Drinfeld have recently
given a notion of ``chiral algebra'' in terms of D-modules on algebraic curves.
This definition consists of a ``skew-symmetry'' relation and a ``Jacobi
identity'' relation in a categorical setting. In this paper, we show directly
that these chiral algebras are essentially the same as vertex algebras without
vacuum vector (and without grading), by establishing an equivalence between the
skew-symmetry and Jacobi identity relations of Beilinson-Drinfeld and the
(similarly-named, but different) skew-symmetry and Jacobi identity relations in
the formal-variable approach to vertex operator algebra theory as formulated by
Borcherds, Frenkel-Lepowsky-Meurman and Frenkel-Huang-Lepowsky.


title: Homological Reduction of Constrained Poisson Algebras

abstract:
  The ``classical BRST construction'' as developed by
Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of
the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson
manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus.
This ideal is called first class if $I$ is closed under the Poisson bracket;
geometers refer to the constraint locus as coisotropic. The physicists' model
is crucially a differential Poisson algebra extension of a Poisson algebra $P$;
its differential contains a piece which reinvented the Koszul complex for the
ideal $I$ and a piece which looks like the Cartan-Chevalley-Eilenberg
differential.
  The present paper is concerned purely with the homological (Poisson)
algebraic structures, using the notion of ``model'' from rational homotopy
theory and the techniques of homological perturbation theory to establish some
of the basic results explaining the mathematical existence of the classical
BRST-BFV construction. Although the usual treatment of BFV is basis dependent
(individual constraints) and nominally finite dimensional, I take care to avoid
assumptions of finite dimensionality and work more invariantly in terms of the
ideal. In particular, the techniques are applied to the `irregular' case (the
ideal is not generated by a regular sequence of constraints), although the
geometric interpretation is less complete.


title:
Algebraic integrability of Macdonald operators and representations of
  quantum groups

abstract:
  In this paper we construct examples of commutative rings of difference
operators with matrix coefficients from representation theory of quantum
groups, generalizing the results of our previous paper to the $q$-deformed
case. A generalized Baker-Akhiezer function $\Psi$ is realized as a matrix
character of a Verma module and is a common eigenfunction for a commutative
ring of difference operators. In particular, we obtain the following result in
Macdonald theory: at integer values of the Macdonald parameter $k$, there exist
difference operators commuting with Macdonald operators which are not
polynomials of Macdonald operators. This result generalizes an analogous result
of Chalyh and Veselov for the case $q=1$, to arbitrary $q$. As a by-product, we
prove a generalized Weyl character formula for Macdonald polynomials (a
conjecture by G.Felder and A.Varchenko), the duality for the $\Psi$-function,
and the existence of shift operators.


title: Interpolation between para-Bose and para-Fermi statistics

abstract:
  Using deformed Green's oscillators and Green's Ansatz,we construct a
multiparameter interpolation between para-Bose and para-Fermi statistics of a
given order. When the interpolating parameters $q_{ij}$ satisfy $|q_{ij}|<1
(|q_{ij}|= 1)$, the interpolation statistics is "infinite quon"-like
(anyon-like).The proposed interpolation does not contain states of negative
norms.


title: Skew Symmetric Bundle Maps on Space-Time

abstract:
  We study the "Lie Algebra" of the group of Gauge Transformations of
Space-time. We obtain topological invariants arising from this Lie Algebra. Our
methods give us fresh mathematical points of view on Lorentz Transformations,
orientation conventions, the Doppler shift, Pauli matrices , Electro-Magnetic
Duality Rotation, Poynting vectors, and the Energy Momentum Tensor $T$ .