title: Semi-Infinite Wedges and Vertex Operators

abstract:
  The level 1 highest weight modules of the quantum affine algebra
$U_q(\widehat{\frak{sl}}_n)$ can be described as spaces of certain
semi-infinite wedges. Using a $q$-antisymmetrization procedure, these
semi-infinite wedges can be realized inside an infinite tensor product of
evaluation modules. This realization gives rise to simple descriptions of
vertex operators and (up to a scalar function) their compositions.


title: Some twisted sectors for the Moonshine Module

abstract:
  For the moonshine module $V^{\natural},$ whose automorphism is the Monster
${\Bbb M},$ We show how to give a uniform existence proof for irreducible
$g$-twisted modules for elements of type $2A,$ $2B$ and $4A$ in ${\Bbb M}.$ The
most interesting of these is the twisted sector $V^{\natural}(2A),$ whose
automorphism group is essentially the centralizer of $2A$ in ${\Bbb M}.$ This
is a 2-fold central extension of the Baby Monster, the second largest sporadic
simple group. We also establish uniqueness of the twisted sectors and the
hauptmodul property for the graded traces of automorphisms of odd order, as
predicted by conformal field theory.


title:
Witten's Invariants of Rational Homology Spheres at Prime Values of $K$
  and Trivial Connection Contribution

abstract:
  We establish a relation between the coefficients of asymptotic expansion of
trivial connection contribution to Witten's invariant of rational homology
spheres and the invariants that T.~Ohtsuki extracted from Witten's invariant at
prime values of $K$. We also rederive the properties of prime $K$ invariants
discovered by H.~Murakami and T.~Ohtsuki. We do this by using the bounds on
Taylor series expansion of the Jones polynomial of algebraically split links,
studied in our previous paper. These bounds are enough to prove that Ohtsuki's
invariants are of finite type. The relation between Ohtsuki's invariants and
trivial connection contribution is verified explicitly for lens spaces and
Seifert manifolds.


title: Path Integrals with Generalized Grassmann Variables

abstract:
  We construct path integral representations for the evolution operator of
q-oscillators with root of unity values of q-parameter using Bargmann-Fock
representations with commuting and non-commuting variables, the differential
calculi being q-deformed in both cases. For $q^2=-1$ we obtain a new form of
Grassmann-like path integral.


title: Introduction to vertex operator algebras I

abstract:
  This is the first part of the revised versions of the notes of three
consecutive expository lectures given by Chongying Dong, Haisheng Li and Yi-Zhi
Huang in the conference on Monster and vertex operator algebras at the Research
Institute of Mathematical Sciences, Kyoto, September 4-9, 1994. In this part we
review the definitions of vertex operator algebras and twisted modules, and
discuss examples.


title: Introduction to vertex operator algebras II

abstract:
  This is the second part of the revised versions of the notes of three
consecutive expository lectures given by Chongying Dong, Haisheng Li and Yi-Zhi
Huang in the conference on Monster and vertex operator algebras at the Research
Institute of Mathematical Sciences, Kyoto, September 4-9, 1994. In this part,
we discuss the commutativity, the associativity, the Jacobi identity and the
relations among them. We also discuss an analogue of endomorphism ring of a
vector space in vertex operator algebra theory.


title: Introduction to vertex operator algebras III

abstract:
  This is the third part of the revised versions of the notes of three
consecutive expository lectures given by Chongying Dong, Haisheng Li and Yi-Zhi
Huang in the conference on Monster and vertex operator algebras at the Research
Institute of Mathematical Sciences, Kyoto, September 4-9, 1994. In this part,
we discuss an $S_{3}$-symmetry of the Jacobi identity, construct the
contragredient module for a module for a vertex operator algebra and apply
these to the construction of the vertex operator map for the moonshine module.
We review the notions of intertwining operator, fusion rule and Verlinde
algebra. We also describe briefly the geometric interpretation of vertex
operator algebras. We end the exposition with an explanation of the role of
vertex operator algebras in conformal field theories.


title: A Link Invariant from Quantum Dilogarithm

abstract:
  The link invariant, arising from the cyclic quantum dilogarithm via the
particular $R$-matrix construction, is proved to coincide with the invariant of
triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A,
Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway
polynomial, vanishes on disjoint union of links. The $R$-matrix can be
considered as the cyclic analog of the universal $R$-matrix associated with
$U_q(sl(2))$ algebra.


title:
Finite-dimensional Calogero representation of the $q$-differential
  operator

abstract:
  A finite-dimensional matrix representation of the Jackson $q$-differential
operator $D_q$, defined by $D_qf(x)$ $=$ $(f(qx)-f(x))/(x(q-1))$, is written
down following Calogero. Such a representation of $D_q$ should have
applications in $q$-analysis leading to the corresponding extensions of the
numerous results of Calogero's work.


title:
Local systems of twisted vertex operators, vertex operator superalgebras
  and twisted modules

abstract:
  We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex
operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion
of local system of vertex operators [Li]. First, we prove that any local system
of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra
structure with an automorphism $\sigma$ of order $T$ with $M$ as a
$\sigma$-twisted module. Then we prove that for a vertex (operator)
superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a
$\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator)
superalgebra homomorphism from $V$ to some local system of
$\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the
twisted modules for vertex operator (super)algebras associated to some
well-known infinite-dimensional Lie (super)algebras and we prove the complete
reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras
associated to standard modules for an affine Lie algebra.


title: Quantum planes and quantum cylinders from Poisson homogeneous spaces

abstract:
  Quantum planes and a new quantum cylinder are obtained as quantization of
Poisson homogeneous spaces of two different Poisson structures on classical
Euclidean group E(2).


title:
Combinatorial constructions of modules for infinite-dimensional Lie
  algebras, II. Parafermionic space

abstract:
  The standard modules for an affine Lie algebra $\ga$ have natural
subquotients called parafermionic spaces -- the underlying spaces for the
so-called parafermionic conformal field theories associated with $\ga.$
  We study the case $\ga = \widehat{sl}(n+1,\C)$ for any positive integral
level $k \geq 2.$ Generalizing the $\cal Z$-algebra approach of Lepowsky,
Wilson and Primc, we construct a combinatorial basis for the parafermionic
spaces in terms of colored partitions. The parts of these partitions represent
''Fourier coefficients'' of generalized vertex operators (parafermionic
currents) and can be interpreted as statistically interacting quasi-particles
of color $i,\;1\leq i \leq n,$ and charge $s,\; 1\leq s \leq k-1.$ From a
combinatorial point of view, these bases are essentially identical with the
bases for level $k-1$ principal subspaces constructed by the author in [GeI].
In the particular case of the vacuum module, the character (string function)
associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS]
conjectured in a Bethe Ansatz layout.
  New combinatorial characters are established for the whole standard vacuum
$\ga$-modules.


title:
Linear representations of groups with translation invariant defining
  relationships. Some new series of braid group representations and new
  invariants of links and knots

abstract:
  In this paper we indicate one method of construction of linear
representations of groups and algebras with translation invariant (except,
maybe , finite number) defining relationships. As an illustration of this
method, we give one approach to the construction of linear representations of
braid group and derive some series of such representations. Some invariants of
oriented knots and links are constructed.
  The author is grateful to Yuri Drozd, Sergey Ovsienko and other members of
algebraic seminar at Kiev University for the creative atmosphere without which
this work could hardly appear.


title:
The theory of physical superselection sectors in terms of vertex
  operator algebra language

abstract:
  We formulate an interpretation of the theory of physical superselection
sectors in terms of vertex operator algebra language. Using this formulation we
give a construction of simple current from a primary semisimple element of
weight one. We then prove that if a rational vertex operator algebra $V$ has a
simple current $M$ satisfying certain conditions, then $V\oplus M$ has a
natural rational vertex operator (super)algebra structure. Applying our results
to a vertex operator algebra associated to an affine Lie algebra, we construct
its simple currents and the extension by a simple current. We also present two
essentially equivalent constructions for twisted modules for an inner
automorphism from the adjoint module or any untwisted module.


title: Bethe Ansatz for higher spin eight vertex models

abstract:
  A generalization of the eight vertex model by means of higher spin
representations of the Sklyanin algebra is investigated by the quantum inverse
scattering method and the algebraic Bethe Ansatz. Under the well-known string
hypothesis low-lying excited states are considered and scattering phase shifts
of two physical particles are calculated. The $S$ matrix of two particle states
is shown to be proportional to the Baxter's elliptic $R$ matrix with a
different elliptic modulus from the original one.


title: On the parametrization of solutions of the Yang--Baxter equations

abstract:
  We study all five-, six-, and one eight-vertex type two-state solutions of
the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13}
A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of
these solution. Starting with algebraic solutions, whose parameters have no
specific interpretation, and then using these symmetries we can construct a
parametrization where we can identify global, color and spectral parameters. We
show in particular how the distribution of these parameters may be changed by a
change of gauge.


title: Gauss Decomposition for Quantum Groups and Supergroups

abstract:
  The Gauss decompositions of the quantum groups, related to classical Lie
groups and supergroups are considered by the elementary algebraic and
$R$-matrix methods. The commutation relations between new basis generators
(which are introduced by the decomposition) are described in some details. It
is shown that the reduction of the independent generator number in the new
basis to the dimension of related classical (super) group is possible. The
classical expression for (super) determinant through the Gauss decomposition
generators is not changed in the deformed case. The symplectic quantum group
$Sp_q(2)$ and supergroups $GL_q(1|1), GL_q(2|1)$ are considered as examples.


title: Cohomology of Drinfel'd algebras: A General Nonsense Approach

abstract:
  In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the
Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a
cohomology controlling deformations of quasi-coassociative (or Drinfel'd)
bialgebras. The full version of the paper will appear as [Markl, Shnider:
Drinfel'd Algebra Deformations, Homotopy Comodules and the Associahedra] in
Trans. Amer. Math. Soc. The construction in the paper was based on very
explicit arguments using deep combinatorial properties of the associahedra. The
present paper gives an alternative, general nonsense approach to the
construction. So, we just prove the existence of such a cohomology without
explicitly constructing it. This should be compared with the two approaches to
the cohomology of associative algebras: we either describe explicitly the
Hochschild complex and say "Behold! this is the cohomology" or we prove the
existence of a projective resolution and define the cohomology as the derived
functor.


title: A Novel Aprroach to Knot Classification

abstract:
  In a previous paper (q-alg/9501022) we suggested some algorithms that could
be useful in solving the problem of knot classification. Here we continue this
discussion by answering questions raised in that paper and by commenting on
practical aspects related to running these algorithms as computer programs.
Further on, we reach a solution by improving on the efficiency of these
algorithms, and by introducing new ones.


title: Relativistic and Newtonian kappa spacetimes

abstract:
  The deformations of the Galilei algebra and their associated noncommutative
Newtonian spacetimes are investigated. This is done by analyzing the possible
nonrelativistic limits of an eleven generator (pseudo)extended \kap-Poincar\'e
algebra $\tilde{\cal P}_\kappa$ and their implications for the existence of a
first order differential calculus. The additional one-form needed to achieve a
consistent calculus on \kap-Minkowski space is shown to be related to the
additional central generator entering in the $\tilde{\cal P}_\kappa$ Hopf
algebra. In the process, deformations of the extended Galilei and Galilei
algebras are introduced which have, respectively, a cocycle and a
bicrossproduct structure.


title:
On quantized algebra of Wess-Zumino differential operators at roots of
  unity

abstract: Quantum algebra of differential operators are studied


title: Recursion relations and branching rules for simple Lie algebras

abstract:
  The branching rules between simple Lie algebras and its regular (maximal)
simple subalgebras are studied. Two types of recursion relations for anomalous
relative multiplicities are obtained. One of them is proved to be the
factorized version of the other. The factorization property is based on the
existence of the set of weights $\Gamma$ specific for each injection. The
structure of $\Gamma$ is easily deduced from the correspondence between the
root systems of algebra and subalgebra. The recursion relations thus obtained
give rise to simple and effective algorithm for branching rules. The details
are exposed by performing the explicit decomposition procedure for $A_{3}
\oplus u(1) \to B_{4}$ injection.


title: The Quantum Galilei Group

abstract:
  The quantum Galilei group $G_{\varkappa}$ is defined. The bicrossproduct
structure of $G_{\varkappa}$ and the corresponding Lie algebra is revealed. The
projective representations for the two-dimensional quantum Galilei group are
constructed.


title: On the spaces of polynomial knots

abstract:
  Homology groups of spaces of nonsingular polynomial embeddings ${\bf R}^1 \to
{\bf R}^n$ of degrees $\le 4$ are calculated. A general algebraic technique of
such calculations for spaces of polynomial knots of arbitrary degrees is
described.

categories: q-alg alg-geom dg-ga math.AG math.DG math.QA


title: New Level-0 Action of $U_q(\widehat{sl}_2)$ on Level-1 Modules

abstract:
  A level-0 action of $U_q(\widehat{sl}_2)$ is defined on the sum of level-1
irreducible highest weight modules. With the aid of the affine Hecke algebras,
this action is realized on the basis created by the vertex operators. This is a
$q$-analogue of the Yangian symmetry in conformal field theory.


abstract:
  This paper is devoted to homological treatment of Harish-Chandra
decomposition for zonal spherical functions of type $A_n$.


title: Cohomology of Quantum Groups

abstract:
  Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie
bialgebras and quantum groups. Contains a new derivation of the classification
of classical r-matrices in terms of deformation cohomology, and a calculation
of the esoteric forms of quantum gl(N) by deformation theory.


title: Elliptic Algebras and Equivariant Elliptic Cohomology

abstract:
  In this paper we explain the parallelism in the classification of three
different kinds of mathematical objects:
  (i) Classical r-matrices.
  (ii) Generalized cohomology theories that have Chern classes for complex
vector bundles.
  (iii) 1-dimensional formal groups.
  The main point of the paper is a construction of the elliptic algebra
associated to Belavin's classical elliptic r-matrix in terms of Equivariant
elliptic cohomology of the Steinberg varieties associated to some partial flag
manifolds.


title: On a q-analogue of the multiple gamma functions

abstract:
  A $q$-analogue of the multiple gamma functions is introduced, and is shown to
satisfy the generalized Bohr-Morellup theorem. Furthermore we give some
expressions of these function.


title: Universal T-matrix for Twisted Quantum gl(N)

abstract:
  The Universal T-matrix is the capstone of the structure that consists of a
quantum group and its dual, and the central object from which spring the
T-matrices (monodromies) of all the associated integrable models. A closed
expression is obtained for the case of multiparameter (twisted) quantum
$gl(N)$. The factorized nature of standard quantum groups, that allows the
explicit expression for $U\hskip-1mm T $ to be obtained with relative ease,
extends to some nonstandard quantum groups, such as those based on $A_n^{(2)}$,
and perhaps to all. The paper is mostly concerned with parameters in general
position, but the extension to roots of unity is also explored, in the case of
$g\ell(N)$. The structure of the dual is now radically different, and an
interesting generalization of the $q$-exponential appears in the formulas for
the Universal T- and R-matrices. The projection to quantum $sl(N)$ is simple
and direct; this allows, in particular, to apply recent results concerning
deformations of twisted $gl(N)$ to the semisimple quotient.


title:
Deformations of quadratic algebras and the corresponding quantum
  semigroups

abstract:
  Let $V$ be a finite dimensional vector space. Given a decomposition $V\otimes
V=\oplus_i^n I_i$, define $n$ quadratic algebras $(V, J_m)$ where
$J_m=\oplus_{i\neq m} I_i$. This decomposition defines also the quantum
semigroup $M(V;I_1,...,I_n)$ which acts on all these quadratic algebras. With
the decomposition we associate a family of associative algebras
$A_k=A_k(I_1,...I_n)$, $k\geq 2$. In the classical case, when $V\otimes V$
decomposes into the symmetric and skewsymmetric tensors, $A_k$ coincides with
the group algebra of the symmetric group $S_k$. Let $I_{ih}$ be deformations of
the subspaces $I_i$. In the paper we give a criteria for flatness of the
corresponding deformations of the quadratic algebras $(V[[h]],J_{ih}$ and the
quantum semigroup $M(V[[h]];I_{1h},...,I_{nh})$. It says that the deformations
will be flat if the algebras $A_k(I_1,...,I_n)$ are semisimple and under the
deformation their dimension does not change.
  Usually, the decomposition into $I_i$ is defined by a given Yang-Baxter
operator $S$ on $V\otimes V$, for which $I_i$ are its eigensubspaces, and the
deformations $I_{ih}$ are defined by a deformation $S_h$ of $S$. We consider
the cases when $S_h$ is a deformation of Hecke or Birman-Wenzl symmetry, and
also the case when $S_h$ is the Yang-Baxter operator which appears by a
representation of the Drinfeld-Jimbo quantum group. Applying the flatness
criteria we prove that in all these cases we obtain flat deformations of the
quadratic algebras and the corresponding quantum semigroups.


title: Deformations of Multiparameter Quantum gl(N)

abstract:
  Multiparameter quantum gl(N) is not a rigid structure. This paper defines an
essential deformation as one that cannot be interpreted in terms of a
similarity transformation, nor as a perturbation of the parameters. All the
equivalence classes of first order essential deformations are found, as well as
a class of exact deformations. This work provides quantization of all the
classical Lie bialgebra structures (constant r-matrices) found by Belavin and
Drinfeld for sl(n). A special case, that requires the Hecke parameter to be a
cubic root of unity, stands out.


title: ISO_q(3) and ISO_q(2,1)

abstract:
  We prove the embedding of $ ISO_q(3) \hook ISU^{ex}_{\sqrt{q}}(2) $ and $
ISO_q(2,1) \hook ISL^{ex}_q(2,R)$ as $^*$-algebras and give a Hilbert space
representation of $ISU^{ex}_{\sqrt{q}}(2)$


title:
A theory of tensor products for module categories for a vertex operator
  algebra, III

abstract:
  This is the third part in a series of papers developing a tensor product
theory for modules for a vertex operator algebra. The goal of this theory is to
construct a ``vertex tensor category'' structure on the category of modules for
a suitable vertex operator algebra. The notion of vertex tensor category is
essentially a ``complex analogue'' of the notion of symmetric tensor category,
and in fact a vertex tensor category produces a braided tensor category in a
natural way. In this paper, we focus on a particular element $P(z)$ of a
certain moduli space of three-punctured Riemann spheres; in general, every
element of this moduli space will give rise to a notion of tensor product, and
one must consider all these notions in order to construct a vertex tensor
category. Here we present the fundamental properties of the $P(z)$-tensor
product of two modules for a vertex operator algebra. We give two constructions
of a $P(z)$-tensor product, using the results, established in Parts I and II of
this series, for a certain other element of the moduli space. The definitions
and results in Part I (hep-th/9309076, which has been replaced by a new version
with a greatly expanded introduction and updated references) and Part II
(hep-th/9309159) are recalled.


title:
A theory of tensor products for module categories for a vertex operator
  algebra, IV

abstract:
  This is the fourth part of a series of papers developing a tensor product
theory of modules for a vertex operator algebra. In this paper, We establish
the associativity of $P(z)$-tensor products for nonzero complex numbers $z$
constructed in Part III of the present series under suitable conditions. The
associativity isomorphisms constructed in this paper are analogous to
associativity isomorphisms for vector space tensor products in the sense that
it relates the tensor products of three elements in three modules taken in
different ways. The main new feature is that they are controlled by the
decompositions of certain spheres with four punctures into spheres with three
punctures using a sewing operation. We also show that under certain conditions,
the existence of the associativity isomorphisms is equivalent to the
associativity (or (nonmeromorphic) operator product expansion in the language
of physicists) for the intertwining operators (or chiral vertex operators).
Thus the associativity of tensor products provides a means to establish the
(nonmeromorphic) operator product expansion.


title:
Virasoro vertex operator algebras, the (nonmeromorphic) operator product
  expansion and the tensor product theory

abstract:
  In the references [HL1]--[HL5] and [H1], a theory of tensor products of
modules for a vertex operator algebra is being developed. To use this theory,
one first has to verify that the vertex operator algebra satisfies certain
conditions. We show in the present paper that for any vertex operator algebra
containing a vertex operator subalgebra isomorphic to a tensor product algebra
of minimal Virasoro vertex operator algebras (vertex operator algebras
associated to minimal models), the tensor product theory can be applied. In
particular, intertwining operators for such a vertex operator algebra satisfy
the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic)
associativity (operator product expansion). Combined with a result announced in
[HL4], the results of the present paper also show that the category of modules
for such a vertex operator algebra has a natural structure of a braided tensor
category. In particular, for any pair $p, q$ of relatively prime positive
integers larger than $1$, the category of minimal modules of central charge
$1-6\frac{(p-q)^{2}}{pq}$ for the Virasoro algebra has a natural structure of a
braided tensor category.


title: Riemannian Geometry on Quantum Spaces

abstract:
  An algebraic formulation of Riemannian geometry on quantum spaces is
presented, where Riemannian metric, distance, Laplacian, connection, and
curvature have their counterparts. This description is also extended to complex
manifolds. Examples include the quantum sphere, the complex quantum projective
spaces and the two-sheeted space.


title: On Hopf algebras and the elimination theorem for free Lie algebras

abstract:
  The elimination theorem for free Lie algebras, a general principle which
describes the structure of a free Lie algebra in terms of free Lie subalgebras,
has been recently used by E. Jurisich to prove that R. Borcherds' ``Monster Lie
algebra'' has certain large free Lie subalgebras, illuminating part of
Borcherds' proof that the moonshine module vertex operator algebra obeys the
Conway-Norton conjectures. In the present expository note, we explain how the
elimination theorem has a very simple and natural generalization to, and
formulation in terms of, Hopf algebras. This fact already follows from general
results contained in unpublished 1972 work, unknown to us when we wrote this
note, of R. Block and P. Leroux.


title: Vector Fields on Quantum Groups

abstract:
  We construct the space of vector fields on a generic quantum group. Its
elements are products of elements of the quantum group itself with left
invariant vector fields. We study the duality between vector fields and 1-forms
and generalize the construction to tensor fields. A Lie derivative along any
(also non left invariant) vector field is proposed and a puzzling ambiguity in
its definition discussed. These results hold for a generic Hopf algebra.


title: Hopf algebroids and quantum groupoids

abstract:
  We introduce the notion of Hopf algebroids, in which neither the total
algebras nor the base algebras are required to be commutative. We give a class
of Hopf algebroids associated to module algebras of the Drinfeld doubles of
Hopf algebras when the $R$-matrices act properly. When this construction is
applied to quantum groups, we get examples of quantum groupoids, which are
semi-classical limits of Poisson groupoids. The example of quantum $sl(2)$ is
worked out in details.


title: Quantum Affine Algebras and Deformations of the Virasoro and W-algebras

abstract:
  We generalize some results concerning affine algebras at the critical level
to the corresponding quantum algebras. In particular, we show that the Wakimoto
realization provides a homomorphism of Poisson algebras from the center of a
quantum affine algebra to a Heisenberg-Poisson algebra. This homomorphism is a
q-deformation of the Miura transformation. It is given by the same formulas as
the spectra of transfer-matrices of the corresponding quantum integrable
models. The image of the center in the Heisenberg-Poisson algebra is a Poisson
subalgebra of the latter, which is a q-deformation of the corresponding
classical W-algebra. The relations in this Poisson algebra are explicitly
computed in the case of sl(n)^.


title: Direct Sum Decompositions and Indecomposable TQFT's

abstract:
  The decomposition of an arbitrary axiomatic topological quantum field theory
or TQFT into indecomposable theories is given. In particular, unitary TQFT's in
arbitrary dimensions are shown to decompose into a sum of theories in which the
Hilbert space of the sphere is one-dimensional, and indecomposable
two-dimensional theories are classified.


title: Killing Form on Quasitriangular Hopf Algebras and Quantum Lie Algebras

abstract:
  The basics of quasitriangular Hopf algebras and quantum Lie algebras are
briefly reviewed, and it is shown that their properties allow the introduction
of a Killing form. For quantum Lie algebras, this leads to the definitions of a
Killing metric and quadratic casimir. The specific case of $\uea{su}{N}$ is
examined in detail, where it is shown that many of the classical results are
reproduced, and explicit calculations to illustrate the conclusions are
presented for $\uea{su}{2}$.


title: On the quantization of Poisson brackets

abstract:
  In this paper we introduce two classes of Poisson brackets on algebras (or on
sheaves of algebras). We call them locally free and nonsingular Poisson
brackets. Using the Fedosov's method we prove that any locally free nonsingular
Poisson bracket can be quantized. In particular, it follows from this that all
Poisson brackets on an arbitrary field of characteristic zero can be quantized.
The well known theorem about the quantization of nondegenerate Poisson brackets
on smooth manifolds follows from the main result of this paper as well.


title: Non-Symmetric Macdonald's Polynomials

abstract:
  We extend the previous paper "Macdonald's evaluation ... and applications" to
the non-symmetric polynomilas recently introduced by Macdonald (as difference
counterparts of Opdam's non-symmetric ones).


title: Lie-Poisson Deformation of the Poincar\'e Algebra

abstract:
  We find a one parameter family of quadratic Poisson structures on ${\bf
R}^4\times SL(2,C)$ which satisfies the property {\it a)} that it is preserved
under the Lie-Poisson action of the Lorentz group, as well as {\it b)} that it
reduces to the standard Poincar\'e algebra for a particular limiting value of
the parameter. (The Lie-Poisson transformations reduce to canonical ones in
that limit, which we therefore refer to as the `canonical limit'.) Like with
the Poincar\'e algebra, our deformed Poincar\'e algebra has two Casimir
functions which we associate with `mass' and `spin'. We parametrize the
symplectic leaves of ${\bf R}^4\times SL(2,C)$ with space-time coordinates,
momenta and spin, thereby obtaining realizations of the deformed algebra for
the cases of a spinless and a spinning particle. The formalism can be applied
for finding a one parameter family of canonically inequivalent descriptions of
the photon.


title:
On Vassiliev knot invariants induced from finite type 3-manifold
  invariants

abstract:
  We prove that the knot invariant induced by a $\Bbb Z$-homology 3-sphere
invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order
$\leq k-2$. The method developed in our computation shows that there is no
$\Bbb Z$-homology 3-sphere invariant of order 5. This result agrees with a
conjecture of Rozansky based on physical predictions about the asymptotic
behavior of Witten's Chern-Simons path integral.


title: Links, Quantum Groups, and TQFT's

abstract:
  The Jones polynomial and the Kauffman bracket are constructed, and their
relation with knot and link theory is described. The quantum groups and tangle
functor formalisms for understanding these invariants and their descendents are
given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial,
is constructed explicitly. The $3$-manifold invariants and the axiomatic
topological quantum field theories which arise from these link invariants at
certain values of the parameter are constructed.


title: Affine Algebras, Langlands Duality and Bethe Ansatz

abstract:
  We review various aspects of representation theory of affine algebras at the
critical level, geometric Langlands correspondence, and Bethe ansatz in the
Gaudin models. Geometric Langlands correspondence relates D-modules on the
moduli space of G-bundles on a complex curve X and flat G^L-bundles on X.
Beilinson and Drinfeld construct it by applying a localization functor to
representations of affine algebras of critical level. We show that in genus
zero the corresponding D-modules are closely related to the diagonalization
problem in the Gaudin model associated to G. This allows us to give a new
interpretation of the Bethe ansatz and Sklyanin's separation of variables in
the Gaudin model in terms of Langlands correspondence.


title: On Deformations of Commutation Relation Algebras

abstract:
  This paper is on $C$ - symmetric creation and annihilation operators, which
are constructed on Wick's algebras which fulfil consistency conditions. The
essential assumption is that every algebraic action must be constant on
equivalence classes. All consistency conditions follow from the above
assumption. In this way we obtain well defined quotient algebras with some
additional relations.


title: Quantization of Lie bialgebras, I

abstract:
  In the paper "On some unsolved problems in quantum group theory", V.Drinfeld
formulated the problem of the existence of a universal quantization for Lie
bialgebras. When the paper "Tensor structures arising from affine Lie algebras,
III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods
could be useful for the problem of universal quantization of Lie bialgebras. In
this paper we use these methods to construct the universal quantization, which
gives a positive answer to Drinfeld's question. We also show the existence of
universal quantization of classical r-matrices, unitary r-matrices, and
quasitriangular Lie bialgebras, which answers the corresponding questions of
Drinfeld.


title: Integral Representations of the Macdonald Symmetric Functions

abstract:
  Multiple-integral representations of the (skew-)Macdonald symmetric functions
are obtained. Some bosonization schemes for the integral representations are
also constructed.


title:
Classification of All Poisson-Lie Structures on an Infinite-Dimensional
  Jet Group

abstract:
  A local classification of all Poisson-Lie structures on an
infinite-dimensional group $G_{\infty}$ of formal power series is given. All
Lie bialgebra structures on the Lie algebra ${\Cal G}_{\infty}$ of $G_{\infty}$
are also classified.


title:
Poisson-Lie Structures on Infinite-Dimensional Jet Groups and Quantum
  Groups Related to Them

abstract:
  We study the problem of classifying all Poisson-Lie structures on the group
$G_{\infty}$ of formal diffeomorphisms of the real line $\zR^{1}$ which leave
the origin fixed, as well as the extended group of diffeomorphisms
$G_{0\infty}\supset G_{\infty}$ whose action on $\zR^{1}$ does not necessarily
fix the origin. A complete local classification of all Poisson-Lie structures
on the groups $G_{\infty}$ and $G_{0\infty}$ is given. This includes a
classification of all Lie-bialgebra structures on the Lie algebra $\Cal
G_{\infty}$ of $G_{\infty}$, which we prove to be all of coboundary type, and a
classification of all Lie-bialgebra strucutures on the Lie algebra $\Cal
G_{0\infty}$ (the Witt algebra) of $G_{0\infty}$ which also turned out to be
all of coboundary type. A large class of Poisson structures on the space
$V_{\lambda}$ of $\lambda$-densities on the real line is found such that
$V_{\lambda}$ becomes a homogeneous Poisson space under the action of the
Poisson-Lie group $G_{\infty}$. We construct a series of quantum semigroups
whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of
$G_{\infty}$ and $G_{0\infty}$.


title: Vassiliev Invariants for Torus Knots

abstract:
  Vassiliev invariants up to order six for arbitrary torus knots $\{ n , m \}$,
with $n$ and $m$ coprime integers, are computed. These invariants are
polynomials in $n$ and $m$ whose degree coincide with their order. Furthermore,
they turn out to be integer-valued in a normalization previously proposed by
the authors.


title:
Differential Calculi of Poincare-Birkhoff-Witt type on Universal
  Enveloping Algebras

abstract:
  Differential calculi of Poincare-Birkhoff-Witt type on universal enveloping
algebras of Lie algebras g are defined. This definition turns out to be
independent of the basis chosen in g. The role of automorphisms of g is
explained. It is proved that no differential calculus of Poincare-Birkhoff-Witt
type exists on semi-simple Lie algebras. Examples are given, namely gl_n,
Abelian Lie algebras, the Heisenberg algebra, the Witt and the Virasoro
algebra. Completely treated are the 2-dimensional solvable Lie algebra, and the
3-dimensional Heisenberg algebra.


title: Localization of $\frak{u}$-modules. IV. Localization on $\Bbb{P}^1$

abstract:
  This article is a sequel to hep-th/9411050, q-alg/9412017, q-alg/9503013.
Given a collection of $m$ finite factorizable sheaves $\{\CX_k\}$, we construct
here some perverse sheaves over configuration spaces of points on a projective
line $\BP^1$ with $m$ additional marked points. We announce here (with sketch
proof) the computation of the cohomology spaces of these sheaves. They turn out
to coincide with certain "semiinfinite" $\Tor$ spaces of the corresponding
$\fu$-modules. As a corollary, we get a description of local systems of
conformal blocks in WZW models in genus $0$ (cf. ~\cite{ms}) as natural
subquotients of some semisimple local systems of geometric origin. In
particular, these local systems are semisimple themselves.


title: On Finite Type 3-Manifold Invariants II

abstract:
  This paper continues the study of finite-type invariants of homology spheres
studied by Ohtsuki and Garoufalidis. We apply the surgery classification of
links to give a diagrammatic description, using ideas of Ohtsuki. This uses a
computation of the surgery equivalence classes of pure braids. We show that the
order of any invariant, in Ohtsukis sense, is a multiple of 3. We also study
the relation between the order of an invariant and that of the knot invariant
it defines.


title: Cohomological construction of quantized universal enveloping algebras

abstract:
  Given an associative algebra $A$, and the category, $\cC$, of its finite
dimensional modules, additional structures on the algebra $A$ induce
corresponding ones on the category $\cC$. Thus, the structure of a rigid
quasi-tensor (braided monoidal) category on $Rep_A$ is induced by an algebra
homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to
conjugation by $\Phi\in A^{\otimes 3}$ (associativity constraint) and
cocommutative up to conjugation by $\cR\in A^{\otimes 2}$ (commutativity
constraint), together with an antiautomorphism (antipode), $S$, of $A$
satisfying the certain compatibility conditions. A morphism of quasi-tensor
structures is given by an element $F\in A^{\otimes 2}$ with suitable induced
actions on $\Phi$, $\cR$ and $S$. Drinfeld defined such a structure on
$A=U(\cG)[[h]]$ for any semisimple Lie algebra $\cG$ with the usual
comultiplication and antipode but nontrivial $\cR$ and $\Phi$ and proved that
the corresponding quasi-tensor category is isomomorphic to the category of
representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra
(QUE), $U_h(\cG)$.
  In the paper we give a direct cohomological construction of the $F$ which
reduces $\Phi$ to the trivial associativity constraint, without any assumption
on the prior existence of a strictly coassociative QUE. Thus we get a new
approach to the DJ quantization. We prove that $F$ can be chosen to satisfy
some additional invariance conditions under (anti)automorphisms of
$U(\cG)[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor
categories. Moreover, we prove that for pure imaginary values of the
deformation parameter, the elements $F$, $R$ and $\Phi$ can be chosen to be


title: q-Gauge Theory

abstract:
  We examine some issues that arise in the q-deformation of a gauge theory. If
the deformation is carried out by replacing the equal time commutators of free
fields by the corresponding q-commutators, the resulting propagators are not
very much different from those of the undeformed theory as long as one is
dealing with weak fields; but the theory still violates causality. If one
postulates a q-deformed S matrix, the corresponding q-causal commutator has 2
poles of different strength and the result again amounts to a deformation of
the Lorentz group.


title: Nonstandard Quantum Groups and Superisation

abstract:
  We obtain the universal R-matrix of the non-standard quantum group associated
to the Alexander-Conway knot polynomial. We show further that this non-standard
quantum group is related to the super-quantum group $U_qgl(1|1)$ by a general
process of superization, which we describe. We also study a twisted variant of
this non-standard quantum group and obtain, as a result, a twisted version of
$U_qgl(1|1)$ as a $q$-supersymmetry of the exterior differential calculus of
any quantum plane of Hecke type, acting by mixing the bosonic $x_i$
co-ordinates and the forms $d x_i$.


title:
Level-0 structure of level-1 $U_q(\widehat{sl}_2)$-modules and Macdonald
  polynomials

abstract:
  The level-$1$ integrable highest weight modules of $U_q(\widehat{sl}_2)$
admit a level-$0$ action of the same algebra. This action is defined using the
affine Hecke algebra and the basis of the level-$1$ module generated by
components of vertex operators. Each level-$1$ module is a direct sum of
finite-dimensional irreducible level-$0$ modules, whose highest weight vector
is expressed in terms of Macdonald polynomials. This decomposition leads to the
fermionic character formula for the level-$1$ modules.


title: On Quantum Lie Algebras and Quantum Root Systems

abstract:
  As a natural generalization of ordinary Lie algebras we introduce the concept
of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain
adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a
quantum Lie bracket given by the quantum adjoint action. The structure
constants of these algebras depend on the quantum deformation parameter $q$ and
they go over into the usual Lie algebras when $q=1$.
  The notions of q-conjugation and q-linearity are introduced. q-linear
analogues of the classical antipode and Cartan involution are defined and a
generalised Killing form, q-linear in the first entry and linear in the second,
is obtained. These structures allow the derivation of symmetries between the
structure constants of quantum Lie algebras.
  The explicitly worked out examples of $g=sl_3$ and $so_5$ illustrate the
results.


title: On the Cremmer-Gervais quantizations of SL(n)

abstract:
  Non-standard quantum groups $C_R [GL(n)]$ and $C_R [SL(n)]$ are constructed
for a two parameter version of the Cremmer-Gervais $R$-matrix. An epimorphism
is constructed from $C_R [GL(n)]$ onto the restricted dual
$U_{\bar{R}}(\frak{gl}(n-1))$ associated to a related smaller $R$-matrix of the
same form.
  A related result is proved concerning factorizable Lie bialgebras. For any
such Lie bialgebra, the dual Lie bialgebra has a canonical homomorphic image
which is again factorizable.


title: Eigenvalus of Casimir Invariants for Type-I Quantum Superalgebras

abstract:
  We present the eigenvalues of the Casimir invariants for the type I quantum
superalgebras on any irreducible highest weight module.


title: Integration on quantum Euclidean space and sphere in $N$ dimensions

abstract:
  Invariant integrals of functions and forms over $q$ - deformed Euclidean
space and spheres in $N$ dimensions are defined and shown to be positive
definite, compatible with the star - structure and to satisfy a cyclic property
involving the $D$ - matrix of $SO_q(N)$. The definition is more general than
the Gaussian integral known so far. Stokes theorem is proved with and without
spherical boundary terms, as well as on the sphere.


title: Reduced Schur Functions and the Littlewood-Richardson Coefficients

abstract:
  The reduced Schur functions are studied. Their relations to the basic
representation of $A^(1)_{r-1}$ and modular representations of the symmetric
groups are clarified. Littlewood-Richardson coefficients appear in the linear
relations among reduced Schur functions.


title: Intertwining operators and Hirota bilinear equations

abstract:
  An interpretation of Hirota bilinear relations for classical $\tau$ functions
is given in terms of intertwining operators. Noncommutative example of
$U_q(sl_2)$ is presented.


title:
Crystallized structure for level 0 part of modified quantum affine
  algebra

abstract:
  Crystal base of the level 0 part of the modified quantum affine algebra
$\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions,
extremal vectors and crystal structure of all irreducible components are
described explicitly.


title:
Type-I Quantum Superalgebras, $q$-Supertrace and Two-variable Link
  Polynomials

abstract:
  A new general eigenvalue formula for the eigenvalues of Casimir invariants,
for the type-I quantum superalgebras, is applied to the construction of link
polynomials associated with {\em any} finite dimensional unitary irrep for
these algebras. This affords a systematic construction of new two-variable link
polynomials asociated with any finite dimensional irrep (with a real highest
weight) for the type-I quantum superalgebras. In particular infinite families
of non-equivalent two-variable link polynomials are determined in fully
explicit form.


title: Classical Limits, Quantum Duality and Lie-Poisson Structures

abstract:
  Quantum duality principle is applied to study classical limits of quantum
algebras and groups. For a certain type of Hopf algebras the explicit procedure
to construct both classical limits is presented. The canonical forms of
quantized Lie-bialgebras are proved to be two-parametric varieties with two
classical limits called dual. When considered from the point of view of
quantized symmetries such varieties can have boundaries that are noncommutative
and noncocommutative. In this case the quantum duality and dual limits still
exist while instead of Lie bialgebra one has a pair of tangent vector fields.
The properties of these constructions called quantizations of Hopf pairs are
studied and illustrated on examples.


title: Schur duality in the toroidal setting

abstract:
  The goal of the present paper is to prove with simple algebraic methods a
Schur duality between Cherednik's double affine Hecke algebra of type GL(l) and
the toroidal quantum group of type SL(n+1) introduced by V. Ginzburg, M.
Kapranov and E. Vasserot in their "Langlands duality on algebraic surfaces".


title: On Some Generalizations of Batalin-Vilkovisky Algebras

abstract:
  We define the concept of higher order differential operators on a general
noncommutative, nonassociative superalgebra A, and show that a vertex operator
superalgebra has plenty of them, namely modes of vertex operators. A linear
operator \Delta is a differential operator of order at most r if an inductively
defined (r+1)-form \Phi_{\Delta}^{r+1} is identically zero. When A is
supercommutative and associative and \Delta is an odd, square zero, second
order differential operator on A, \Phi_{\Delta}^2 defines a "Batalin-Vilkovisky
algebra" structure on A. We generalize this notion to any superalgebra with an
odd, square zero, second order differential operator, and show that all
properties of the classical BV bracket but one continue to hold "on the nose".
We also point out connections to Leibniz algebras and the noncommutative
homology theory of Loday. Taking the generalization process one step further,
we remove all conditions on \Delta to examine the changes in the basic
properties of the bracket. We see that a topological chiral algebra with a mild
restriction yields a classical BV algebra in the cohomology. Finally, we
investigate the quantum BV master equation and relate it to deformations of
differential graded algebras.


title:
The q-Euclidean algebra $U_q(e^N)$ and the corresponding q-Euclidean
  lattice

abstract:
  We review the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross
SO_{q^{-1}}(N))$ and describe its fundamental Hilbert space representations
\cite{fioeu}, which turn out to be rather simple "lattice-regularized" versions
of the classical ones, in the sense that the spectra of squared momentum
components are discrete and the corresponding eigenfunctions normalizable.These
representations can be regarded as describing a quantum system consisting of
one free particle on the quantum Euclidean space. A suitable notion of
classical limit is introduced, so that we recover the classical continuous
spectra and generalized (non-normalizable) eigenfunctions in that limit.


title:
Link Invariants and Combinatorial Quantization of Hamiltonian
  Chern-Simons Theory

abstract:
  We define and study the properties of observables associated to any link in
$\Sigma\times {\bf R}$ (where $\Sigma$ is a compact surface) using the
combinatorial quantization of hamiltonian Chern-Simons theory. These
observables are traces of holonomies in a non commutative Yang-Mills theory
where the gauge symmetry is ensured by a quantum group. We show that these
observables are link invariants taking values in a non commutative algebra, the
so called Moduli Algebra. When $\Sigma=S^2$ these link invariants are pure
numbers and are equal to Reshetikhin-Turaev link invariants.


title: On The non-commutative Riemannian geometry of GL_q(n)

abstract:
  A recently proposed definition of a linear connection in non-commutative
geometry, based on a generalized permutation, is used to construct linear
connections on GL_q(n). Restrictions on the generalized permutation arising
from the stability of linear connections under involution are discussed.
Candidates for generalized permutation on GL_q(n) are found. It is shown that,
for a given generalized permutation, there exists one and only one associated
linear connection. Properties of the linear connection are discussed, in
particular its bicovariance, torsion and commutative limit.


title: Bethe Subalgebras in Twisted Yangians

abstract:
  We study analogues of the Yangian of the Lie algebra $gl_N$ for the other
classical Lie algebras $so_N$ and $sp_N$. We call them twisted Yangians. They
are coideal subalgebras in the Yangian $Y(gl_N)$ of $gl_N$ and admit
homomorphisms onto the universal enveloping algebras $U(so_N)$ and $U(sp_N)$
respectively. In every twisted Yangian we construct a family of maximal
commutative subalgebras parametrized by the regular semisimple elements of the
corresponding classical Lie algebra. The images in $U(so_N)$ and $U(sp_N)$ of
these subalgebras are also maximal commutative.


title: Quantum Double and Differential Calculi

abstract:
  We show that bicovariant bimodules as defined by Woronowicz are in one to one
correspondence with the Drinfeld quantum double representations. We then prove
that a differential calculus associated to a bicovariant bimodule of dimension
n is connected to the existence of a particular (n+1)--dimensional
representation of the double. An example of bicovariant differential calculus
on the non quasitriangular quantum group E_q(2) is developed. The construction
is studied in terms of Hochschild cohomology and a correspondence between
differential calculi and 1-cocycles is proved. Some differences of calculi on
quantum and finite groups with respect to Lie groups are stressed.


title:
Quantum Deformations of Conformal Algebras Introducing Fundamental Mass
  Parameters

abstract:
  We consider new class of classical r-matrices for D=3 and D=4 conformal Lie
algebras. These r-matrices do satisfy the classical Yang-Baxter equation and as
two-tensors belong to the tensor product of Borel subalgebra. In such a way we
generalize the lowest order of known nonstandard quantum deformation of sl(2)
to the Lie algebras sp(4) = so(5) and sl(4) = so(6). As an exercise we
interpret nonstandard deformation of sl(2) as describing quantum D=1 conformal
algebra with fundamental mass parameter. Further we describe the D=3 and D=4
conformal bialgebras with deformation parameters equal to the inverse of
fundamental masses. It appears that for D=4 the deformation of the Poincare
algebra sector coincides with "null plane" quantum Poincare algebra.


title: 4-Dimensional BF Theory as a Topological Quantum Field Theory

abstract:
  Starting from a Lie group G whose Lie algebra is equipped with an invariant
nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory
with cosmological term gives rise to a TQFT satisfying a generalization of
Atiyah's axioms to manifolds equipped with principal G-bundle. The case G =
GL(4,R) is especially interesting because every 4-manifold is then naturally
equipped with a principal G-bundle, namely its frame bundle. In this case, the
partition function of a compact oriented 4-manifold is the exponential of its
signature, and the resulting TQFT is isomorphic to that constructed by Crane
and Yetter using a state sum model, or by Broda using a surgery presentation of
4-manifolds.


title:
Geometry of Quantum Group Twists, Multidimensional Jackson Calculus and
  Regularization

abstract:
  We show that R-matricies of all simple quantum groups have the properties
which permit to present quantum group twists as transitions to other coordinate
frames on quantum spaces. This implies physical equivalence of field theories
invariant with respect to q-groups (considered as q-deformed space-time groups
of transformations) connected with each other by the twists. Taking into
account this freedom we study quantum spaces of the special type: with
commuting coordinates but with q-deformed differential calculus and construct
$GL_r(N)$ invariant multidimensional Jackson derivatives. We consider a
particle and field theory on a two-dimensional q-space of this kind and come to
the conclusion that only one (time-like) coordinate proved to be discretized.


title: On a Deformation of $sl(2)$ with Paragrassmannian Variables

abstract:
  We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is
realized by multiplying $\delta$ ($q=e^{\delta}$, $\delta\in \CC$) by $\theta$,
where $\theta$ is a real nilpotent -paragrassmannian- variable of order $r$
($\theta^{r+1}=0$) that we call the order of deformation, the limit
$r\rightarrow \infty$ giving back the standard ${\cal U}_{\displaystyle
{q}}(sl(2))$. In particular we show that, for $r=1$, there exists a new ${\cal
R}$-matrix associated with $sl(2)$. We also proof that the restriction of the
values of the parameters of deformation give nonlinear algebras as particular
cases.


title: The exponential map for representations of $U_{p,q}(gl(2))$

abstract:
  For the quantum group $GL_{p,q}(2)$ and the corresponding quantum algebra
$U_{p,q}(gl(2))$ Fronsdal and Galindo explicitly constructed the so-called
universal $T$-matrix. In a previous paper we showed how this universal
$T$-matrix can be used to exponentiate representations from the quantum algebra
to get representations (left comodules) for the quantum group. Here, further
properties of the universal $T$-matrix are illustrated. In particular, it is
shown how to obtain comodules of the quantum algebra by exponentiating modules
of the quantum group. Also the relation with the universal $R$-matrix is
discussed.


title: Noncommutative Vieta Theorem and Symmetric Functions

abstract:
  A version of the classical Vieta theorem for free noncommuting variables is
given. It leads to a new start in a construction of noncommutative symmetric
functions


title: A generalization of Selberg integral

abstract:
  We analyze the situation which is related to zonal spherical functions of
type $A_n$ and obtain a generalization of Selberg integral.


title: Some Remarks on Producing Hopf Algebras

abstract:
  We report some observations concerning two well-known approaches to
construction of quantum groups. Thus, starting from a bialgebra of
inhomogeneous type and imposing quadratic, cubic or quartic commutation
relations on a subset of its generators we come, in each case, to a q-deformed
universal enveloping algebra of a certain simple Lie algebra. An interesting
correlation between the order of initial commutation relations and the Cartan
matrix of the resulting algebra is observed. Another example demonstrates that
the bialgebra structure of sl_q(2) can be completely determined by requiring
the q-oscillator algebra to be its covariant comodule, in analogy with Manin's
approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic
quantum planes.


title: The Braided Quantum 2-Sphere

abstract:
  In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum
complex manifold. Here we consider several copies of $S^2_q$ and derive their
braiding commutation relations. The braiding is extended to the differential
and to the integral calculus on the spheres. A quantum analogue of the
classical anharmonic ratio of four points on the sphere is given, which is
invariant under the coaction of $SU_q(2)$.


title:
A proof of polynomial identities of type $\widehat{sl(n)_1} \otimes
  \widehat{sl(n)_1} / \widehat{sl(n)_2}$

abstract:
  We present a proof of polynomial identities related to finite analogues of
the branching functions of the coset $\widehat{sl(n)_1} \otimes
\widehat{sl(n)_1} / \widehat{sl(n)_2}$.


title: A TQFT for Wormhole cobordisms over the field of rational functions

abstract:
  We consider a cobordism category whose morphisms are punctured connect sums
of $S^1 \times S^2$'s (wormhole spaces) with embedded admissibly colored banded
trivalent graphs. We define a TQFT on this cobordism category over the field of
rational functions in an indeterminant $A.$ For $r$ large, we recover, by
specializing $A$ to a primitive 4rth root of unity, the
Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus,
for $r$ large, the $r$th Witten-Reshetikhin-Turaev invariant of a link in some
wormhole space, properly normalized, is the value of a certain rational
function at $e^{\frac{\pi i}{2r}}.$ We relate our work to Hoste and Przytycki's
calculation of the Kauffman bracket skein module of $S^1 \times S^2.$


title: Braided Clifford Algebras as Braided Quantum Groups

abstract:
  The paper deals with braided Clifford algebras, understood as
Chevalley-Kahler deformations of braided exterior algebras. It is shown that
Clifford algebras based on involutive braids can be naturally endowed with a
braided quantum group structure. Basic group entities are constructed
explicitly.


title: Characteristic Classes of Quantum Principal Bundles

abstract:
  A noncommutative-geometric generalization of classical Weil theory of
characteristic classes is presented, in the conceptual framework of quantum
principal bundles. A particular care is given to the case when the bundle does
not admit regular connections. A cohomological description of the domain of the
Weil homomorphism is given. Relations between universal characteristic classes
for the regular and the general case are analyzed. In analogy with classical
geometry, a natural spectral sequence is introduced and investigated. The
appropriate counterpart of the Chern character is constructed, for structures
admitting regular connections. Illustrative examples and constructions are
presented.


title: Quantum Principal Bundles and Tannaka-Krein Duality Theory

abstract:
  The structure of quantum principal bundles is studied, from the viewpoint of
Tannaka-Krein duality theory. It is shown that if the structure quantum group
is compact, principal G-bundles over a quantum space M are in a natural
correspondence with certain contravariant functors defined on the category of
finite-dimensional unitary representations of G, with the values in the
category of finite projective bimodules over a *-algebra representing the base
space.


title: Geometry of Quantum Principal Bundles I

abstract:
  A theory of principal bundles possessing quantum structure groups and
classical base manifolds is presented. Structural analysis of such quantum
principal bundles is performed. A differential calculus is constructed,
combining differential forms on the base manifold with an appropriate
differential calculus on the structure quantum group. Relations between the
calculus on the group and the calculus on the bundle are investigated. A
concept of (pseudo)tensoriality is formulated. The formalism of connections is
developed. In particular, operators of horizontal projection, covariant
derivative and curvature are constructed and analyzed. Generalizations of the
first structure equation and of the Bianchi identity are found. Illustrative
examples are presented.


title: On Framed Quantum Principal Bundles

abstract:
  A noncommutative-geometric formalism of framed principal bundles is sketched,
in a special case of quantum bundles (over quantum spaces) possessing classical
structure groups. Quantum counterparts of torsion operators and Levi-Civita
type connections are analyzed. A construction of a natural differential
calculus on framed bundles is described. Illustrative examples are presented.


title: Quantum Principal Bundles and Corresponding Gauge Theories

abstract:
  A generalization of classical gauge theory is presented, in the framework of
a noncommutative-geometric formalism of quantum principal bundles over smooth
manifolds. Quantum counterparts of classical gauge bundles, and classical gauge
transformations, are introduced and investigated. A natural differential
calculus on quantum gauge bundles is constructed and analyzed. Kinematical and
dynamical properties of corresponding gauge theories are discussed.


title: Quantum Principal Bundles as Hopf-Galois Extensions

abstract:
  It is shown that every quantum principal bundle with a compact structure
group is a Hopf-Galois extension. This property naturally extends to the level
of general differential structures, so that every differential calculus over a
quantum principal bundle with a compact structure group is a
graded-differential variant of the Hopf-Galois extension.


title: Functional Integration on Spaces of Connections

abstract:
  Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal
$G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections
on $P$ be a continuous function of the holonomies of $A$ along finitely many
piecewise smoothly immersed curves in $M$, and let a generalized measure on
$\A$ be a bounded linear functional on cylinder functions. We construct a
generalized measure on the space of connections that extends the uniform
measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is
invariant under all automorphisms of $P$, not necessarily the identity on the
base space $M$. Using `spin networks' we construct explicit functions spanning
the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge
transformations.


title: Quantum Deformations of Multi-Instanton Solutions in the Twistor Space

abstract:
  We consider the quantum-group self-duality equation in the framework of the
gauge theory on a deformed twistor space. Quantum deformation of the
Atiyah-Drinfel'd-Hitchin-Manin and t'Hooft multi-instanton solutions are
constructed.


title: Spectral Decomposition of Path Space in Solvable Lattice Model

abstract:
  We give the {\it spectral decomposition} of the path space of the
$U_q(\hatsl)$ vertex model with respect to the local energy functions. The
result suggests the hidden Yangian module structure on the $\hatsl$ level $l$
integrable modules, which is consistent with the earlier work [1] in the level
one case. Also we prove the fermionic character formula of the $\hatsl$ level
$l$ integrable representations in consequence.


title:
Unitarizable Representations of the Deformed Para-Bose Superalgebra
  Uq[osp(1/2)] at Roots of 1

abstract:
  The unitarizable irreps of the deformed para-Bose superalgebra $pB_q$, which
is isomorphic to $U_q[osp(1/2)]$, are classified at $q$ being root of 1. New
finite-dimensional irreps of $U_q[osp(1/2)]$ are found. Explicit expressions
for the matrix elements are written down.


title:
New Solutions of the Yang-Baxter Equation Based on Root of 1
  Representations of the Para-Bose Superalgebra U$_q$[osp(1/2)]

abstract:
  New solutions of the quantum Yang-Baxter equation, depending in general on
three arbitrary parameters, are written down. They are based on the root of
unity representations of the quantum orthosymplectic superalgebra \\U, which
were found recently. Representations of the braid group $B_N$ are defined
within any $N^{th}$ tensorial power of root of 1 \\U modules.


title: On the first order operators in bimodules

abstract:
  We analyse the structure of the first order operators in bimodules introduced
by A. Connes. We apply this analysis to the theory of connections on bimodules
generalizing thereby several proposals.


title: The gl(M|N) Super Yangian and Its Finite Dimensional Representations

abstract:
  Methods are developed for systematically constructing the finite dimensional
irreducible representations of the super Yangian Y(gl(M|N)) associated with the
Lie superalgebra gl(M|N). It is also shown that every finite dimensional
irreducible representation of Y(gl(M|N)) is of highest weight type, and is
uniquely characterized by a highest weight. The necessary and sufficient
conditions for an irrep to be finite dimensional are given.


title: Submanifolds and Quotient Manifolds in Noncommutative Geometry

abstract:
  We define and study noncommutative generalizations of submanifolds and
quotient manifolds, for the derivation-based differential calculus introduced
by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these
definitions.


title:
A Relation Between the Kauffman and the HOMFLY Polynomials for Torus
  Knots

abstract:
  Polynomial invariants corresponding to the fundamental representation of the
gauge group $SO(N)$ are computed for arbitrary torus knots in the framework of
Chern-Simons gauge theory making use of knot operators. As a result, a formula
which relates the Kauffman and the HOMFLY polynomials for torus knots is
presented.


title: Linear Connections on the Two Parameter Quantum Plane

abstract:
  We apply a recently proposed definition of a linear connection in non
commutative geometry based on the natural bimodule structure of the algebra of
differential forms to the case of the two-parameter quantum plane. We find that
there exists a non trivial family of linear connections only when the two
parameters obeys a specific relation.


title: Self-dual Koornwinder-Macdonald polynomials

abstract:
  We prove certain duality properties and present recurrence relations for a
four-parameter family of self-dual Koornwinder-Macdonald polynomials. The
recurrence relations are used to verify Macdonald's normalization conjectures
for these polynomials.


title:
A Quantum Deformation of the Virasoro Algebra and the Macdonald
  Symmetric Functions

abstract:
  A quantum deformation of the Virasoro algebra is defined. The Kac
determinants at arbitrary levels are conjectured. We construct a bosonic
realization of the quantum deformed Virasoro algebra. Singular vectors are
expressed by the Macdonald symmetric functions. This is proved by constructing
screening currents acting on the bosonic Fock space.


title: On Finiteness of Certain Vassiliev Invariants

abstract:
  The best known examples of Vassiliev invariants are the coefficients of a
Jones-type polynomial expanded after exponential substitution. We show that for
a given knot, the first $N$ Vassiliev invariants in this family determine the
rest for some integer $N$.


title: Quantized Lax Equations and Their Solutions

abstract:
  Integrable systems on quantum groups are investigated. The Heisenberg
equations possessing the Lax form are solved in terms of the solution to the
factorization problem on the corresponding quantum group.


title: Scattering matrices and affine Hecke algebras

abstract:
  We construct the scattering matrices for an arbitrary Weyl group in terms of
elementary operators which obey the generalised Yang-Baxter equation. We use
this construction to obtain the affine Hecke algebras. The center of the affine
Hecke algebras coincides with commuting Hamiltonians. These Hamiltonians have
q-deformed affine Lie algebras as symmetry algebra.


title: Toda Fields on Riemann Surfaces: remarks on the Miura transformation

abstract:
  We point out that the Miura transformation is related to a holomorphic
foliation in a relative flag manifold over a Riemann Surface. Certain
differential operators corresponding to a free field description of
$W$--algebras are thus interpreted as partial connections associated to the
foliation.


title: Yangians: their Representations and Characters

abstract:
  We give several formulas for the character of an arbitrary irreducible
finite--dimensional representation for the Yangian of sl_2.


title: Quantum categories. Quantization of the category of linear spaces

abstract:
  We define generalized bialgebras and Hopf algebras and on this basis we
introduce quantum categories and quantum groupoids. The quantization of the
category of linear (super)spaces is constructed. We establish a criterion for
the classical value of the dimension of the polynomial function algebra on the
full subcategory of this quantum category specified by Sudbery type commutation
relations for quantum vector spaces (the ``Poincare -- Birkhoff -- Witt
property''). The criterion is the equality of the ``quantum constants''
$c_\a=c_\b^{\pm 1}$ of the quantum vector spaces. (These constants appeared in
earlier works on PBW for the multiparameter quantization of the general linear
group.) Links with the Yang-Baxter equation and the Yang -- Baxter structures
of quantum linear spaces are established. The role of categories as a
generalization of groups is discussed.


title: Decomposition of $q$-deformed Fock spaces

abstract:
  A decomposition of the level-one $q$-deformed Fock representations of $\uqn$
is given. It is found that the action of $\upqn$ on these Fock spaces is
centralized by a Heisenberg algebra, which arises from the center of the affine
Hecke algebra $\widehat{H}_N$ in the limit $N \rightarrow \infty$. The
$q$-deformed Fock space is shown to be isomorphic as a
$\upqn$-Heisenberg-bimodule to the tensor product of a level-one irreducible
highest weight representation of $\upqn$ and the Fock representation of the
Heisenberg algebra. The isomorphism is used to decompose the $q$-wedging
operators, which are intertwiners between the $q$-deformed Fock spaces, into
constituents coming from $\upqn$ and from the Heisenberg algebra.


title: On the Classification of quantum qroup stuctures on the group GL(2)

abstract:
  All quantum group structures on the group GL(2) are classified. It is shown
that there are only two such structures, the well known quantum groups
GL$_{qp}$(2) and GL$_{hh'}$(2).


title: Quantum Fibre Bundles. An Introduction

abstract:
  An approach to construction of a quantum group gauge theory based on the
quantum group generalisation of fibre bundles is reviewed.


title: Quantum W-algebras and Elliptic Algebras

abstract:
  We define quantum W-algebras generalizing the results of Reshetikhin and the
second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated
to sl_N is an associative algebra depending on two parameters. For special
values of parameters it becomes the ordinary W-algebra of sl_N, or the
q-deformed classical W-algebra of sl_N. We construct free field realizations of
the quantum W-algebras and the screening currents. We also point out some
interesting elliptic structures arising in these algebras. In particular, we
show that the screening currents satisfy elliptic analogues of the Drinfeld
relations in U_q(n^).


title: Generalization of the $h$-Deformation to Higher Dimensions

abstract:
  In this article we construct $GL_{h}(3)$ from $GL_{q}(3)$ by a singular map.
We show that there exist two singular maps which map $GL_{q}(3)$ to new quantum
groups. We also construct their $R$-matrices and will show although the maps
are singular but their $R$-matrices are not. Then we generalize these singular
maps to the case $GL(N)$ and for $C_{n}$ series.


title: Quantum $W_N$ Algebras and Macdonald Polynomials

abstract:
  We derive a quantum deformation of the $W_N$ algebra and its quantum Miura
transformation, whose singular vectors realize the Macdonald polynomials.


title:
Twisted quantum affine algebras and solutions to the Yang-Baxter
  equation

abstract:
  We construct spectral parameter dependent R-matrices for the quantized
enveloping algebras of twisted affine Lie algebras. These give new solutions to
the spectral parameter dependent quantum Yang-Baxter equation.


title: Quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$

abstract:
  Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are
embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo
in the same way as ordinary Lie algebras are embedded into their enveloping
algebras. The quantum Lie product on $\qlie{g}$ is induced by the quantum
adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to
$U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the
quantum root systems, which are now functions of the quantum parameter $q$.
They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.


title: On the deformability of Heisenberg algebras

abstract:
  Based on the vanishing of the second Hochschild cohomology group of the
enveloping algebra of the Heisenberg algebra it is shown that differential
algebras coming from quantum groups do not provide a non-trivial deformation of
quantum mechanics. For the case of a q-oscillator there exists a deforming map
to the classical algebra. It is shown that the differential calculus on quantum
planes with involution, i.e. if one works in position-momentum realization, can
be mapped on a q-difference calculus on a commutative real space. Although this
calculus leads to an interesting discretization it is proved that it can be
realized by generators of the undeformed algebra and does not posess a proper
group of global transformations.


title: Real Forms of the Oscillator Quantum Algebra and its Representations

abstract:
  We consider the conditions under which the $q$-oscillator algebra becomes a
Hopf $*$-algebra. In particular, we show that there are at least two real forms
associated with the algebra. Furthermore, through the representations, it is
shown that they are related to ${\text {su}}_{q^{1/2}}(2)$ with different
conjugations.


title:
Universal R-matrices for finite Abelian groups -- a new look at graded
  multilinear algebra

abstract:
  The universal R-matrices and, dually, the coquasitriangular structures of the
group Hopf algebra of a finite Abelian group (resp. of an arbitrary Abelian
group) are determined. This is used to formulate graded multilinear algebra in
terms of triangular or cotriangular Hopf algebras. For the convenience of the
reader, in a separate section the definitions and basic properties of
quasitriangular and coquasitriangular Hopf algebras are recalled.


title: On inner product in modular tensor categories. I

abstract:
  In this paper we study modular tensor categories (braided rigid balanced
tensor categories with additional finiteness and non-degeneracy conditions), in
particular, representations of quantum groups at roots of unity. We show that
the action of modular group on certain spaces of morphisms in MTC is unitary
with respect to the natural inner product on these spaces. In a special case of
category based on representations of the quantum group U_q sl_n at roots of
unity we show that in some of these spaces of morphisms (for U_q sl_2, in all
of them) the action of modular group can be written in terms of values of
Macdonald's polynomials of type A at roots of unity. This gives identities for
these special values, both known before (symmetry identity) and new ones.
  The paper contains a detailed exposition of the theory of modular categories
as well as construction of modular categories from representation of quantum
groups at roots of unity


title: Regularity of rational vertex operator algebras

abstract:
  A regular vertex operator algebra is a vertex operator algebra such that any
weak module (without grading) is a direct sum of ordinary irreducible modules.
In this paper we give several sufficient conditions under which a rational
vertex operator algebra is regular. We prove that the moonshine module vertex
operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$
associated with the integrable representations of affine algebras of level $l,$
the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest
weight representations for the discrete series of the Virasoro algebra and the
vertex operator algebras $V_L$ associated with positive definite even lattices
$L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable
module of level $l$ for the corresponding affine Lie algebra is a direct sum of
irreducible highest weight integrable modules. The space $V_L$ in general is a
vertex algebra if $L$ is not positive definite. In this case we establish the
complete reducibility of any weak module.


title: Affine Hecke algebras associated to Kac-Moody groups

abstract:
  In this paper we give a geometric construction of Cherednik's double affine
Hecke algebra.
  We construct the algebra as the equivariant $K$-theory of the Lagrangian
subvariety of the cotangent variety of the square of the flag variety of $G$,
the variety being given by the union of the conormal bundles to the $G$-orbits
on $\flag\times\flag$.
  This is a generalisation of work of Kazhdan and Lusztig to the Kac-Moody
case, and is suitable for describing a certain class of modules for this
algebra. In a paper in preparation we will do this, in the case $G$ is affine
(Cherednik's case).


title: Remarks on central extensions of the Galilei group in 2+1 dimensions

abstract:
  Some properties of central extensions of 2+1 dimensional Galilei group are
discussed. It is shown that certain families of extensions are isomorphic. An
interpretation of new nontrivial cocycle is offered. A few bibliographical
remarks are included.


title:
The bicovariant differential calculus on the kappa-Poincare group and on
  the kappa-Minkowski space

abstract:
  The bicovariant differential calculus on the four-dimensional kappa-Poincare
group and the corresponding Lie-algebra like structure are described. The
deifferential calculus on the n-dimensional kappa-Minkowski space covariant
under the action of the kappa-Poincare group is constructed.


title: (T*B)_q, q-analogue of model space and CGC generating matrices

abstract:
  We study relations between the deformed cotangent bundle (T*B)_q for the
Borel subgroup B of a given simple Lie group G, the quantum Lie algebra J_q
associated with the corresponding quantum group G_q and the matrices generating
Clebsch-Gordan coefficients (CGC) for J_q. We reveal the connection of these
objects to quantum analogue of the model space M and q-tensor operators.


title: On the Drinfel'd-Kohno Equivalence of Groups and Quantum Groups

abstract:
  A method to calculate matrix representations of the twist element $\ff$ of
Drinfel'd -- chosen to be unitary -- is given and illustrated at some examples.
It is observed that for these F-matrices the crystal limit $q\!\to\! 0$ exists
and that \mbox{F-matrices} twisting from $0$ to $q$ are of a simpler form than
F-matrices twisting from $1$ to $q$. These results lead to a new interpretation
of $q$-deformation in terms of tensor products of finite-dimensional
representations of compact simple Lie groups.


title:
A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley
  conjecture

abstract:
  A formula of Rodrigues-type for the Jack polynomials is presented. It is seen
to imply a weak form of a conjecture of Macdonald and Stanley.


title: Exact operator solution of the Calogero-Sutherland model

abstract:
  The wave functions of the Calogero-Sutherland model are known to be
expressible in terms of Jack polynomials. A formula which allows to obtain the
wave functions of the excited states by acting with a string of creation
operators on the wave function of the ground state is presented and derived.
The creation operators that enter in this formula of Rodrigues-type for the
Jack polynomials involve Dunkl operators.


title: Noncommutative Geometry of Finite Groups

abstract:
  A finite set can be supplied with a group structure which can then be used to
select (classes of) differential calculi on it via the notions of left-, right-
and bicovariance. A corresponding framework has been developed by Woronowicz,
more generally for Hopf algebras including quantum groups. A differential
calculus is regarded as the most basic structure needed for the introduction of
further geometric notions like linear connections and, moreover, for the
formulation of field theories and dynamics on finite sets. Associated with each
bicovariant first order differential calculus on a finite group is a braid
operator which plays an important role for the construction of distinguished
geometric structures. For a covariant calculus, there are notions of invariance
for linear connections and tensors. All these concepts are explored for finite
groups and illustrated with examples. Some results are formulated more
generally for arbitrary associative (Hopf) algebras. In particular, the problem
of extension of a connection on a bimodule (over an associative algebra) to
tensor products is investigated, leading to the class of `extensible
connections'. It is shown that invariance properties of an extensible
connection on a bimodule over a Hopf algebra are carried over to the extension.
Furthermore, an invariance property of a connection is also shared by a `dual
connection' which exists on the dual bimodule (as defined in this work).


title: Twisted representations of vertex operator algebras

abstract:
  Let $V$ be a vertex operator algebra and $g$ an automorphism of finite order.
We construct an associative algebra $A_g(V)$ and a pair of functors between the
category of $A_g(V)$-modules and a certain category of admissible $g$-twisted
$V$-modules. In particular, these functors exhibit a bijection between the
simple modules in each category. We give various applications, including the
fact that the complete reducibility of admissible $g$-twisted modules implies
both the finite-dimensionality of homogeneous spaces and the finiteness of the
number of simple $g$-twisted modules.


title: Two-Rowed Hecke Algebra Representations at Roots of Unity

abstract:
  In this paper, we initiate a study into the explicit construction of
irreducible representations of the Hecke algebra $H_n(q)$ of type $A_{n-1}$ in
the non-generic case where $q$ is a root of unity. The approach is via the
Specht modules of $H_n(q)$ which are irreducible in the generic case, and
possess a natural basis indexed by Young tableaux. The general framework in
which the irreducible non-generic $H_n(q)$-modules are to be constructed is set
up and, in particular, the full set of modules corresponding to two-part
partitions is described. Plentiful examples are given.


title: A q-Lorentz Algebra From q-Deformed Harmonic Oscillators

abstract:
  A mapping between the operators of the bosonic oscillator and the Lorentz
rotation and boost generators is presented. The analog of this map in the
$q$-deformed regime is then applied to $q$-deformed bosonic oscillators to
generate a $q$-deformed Lorentz algebra, via an inverse of the standard chiral
decomposition. A fundamental representation, and the co-algebra structure, are
given, and the generators are reformulated into $q$-deformed rotations and
boosts. Finally, a relation between the $q$-boson operators and a basis of
$q$-deformed Minkowski coordinates is noted.


title: A Coassociative C*-Quantum Group with Non-Integral Dimensions

abstract:
  By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing
for the coassociative coproduct to be non-unital we obtain a quantum group,
that we call a weak C*-Hopf algebra, which is sufficiently general to describe
the symmetries of essentially arbitrary fusion rules. This amounts to
generalizing the Baaj-Skandalis multiplicative unitaries to multipicative
partial isometries. Every weak C*-Hopf algebra has a dual which is again a weak
C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules.
We shortly discuss applications to amalgamated crossed products, doubles, and
quantum chains.


title: On Ohtsuki's invariants of integral homology 3-spheres, I

abstract:
  An attempt is made to conceptualize the derivation as well as to facilitate
the computation of Ohtsuki's rational invariants $\lambda_n$ of integral
homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants.
Several interesting consequences will follow from our computation of
$\lambda_2$. One of them says that $\lambda_2$ is always an integer divisible
by 3. It seems interesting to compare this result with the fact shown by
Murakami that $\lambda_{1}$ is 6 times the Casson invariant. Other consequences
include some general criteria for distinguishing homology 3-spheres obtained
from surgery on knots by using the Jones polynomial.


title: The HOMFLY polynomial for links in rational homology 3-spheres

abstract:
  We construct a polynomial invariant, for links in a Seifert fibered or
atoroidal rational homology 3-sphere, which generalizes the 2-variable Jones
polynomial (HOMFLY). As a consequence, we show that the dual of the HOMFLY
skein module of a homotopy 3-sphere is isomorpic to that of the genuine
3-sphere .


title:
Non--Minimal $q$--Deformations and Orthogonal Symmetries: $U_q$(SO(5))
  Example

abstract:
  Non--minimal $q$-deformations are defined. Their role in the explicit
construction of the matrix elements of the generators of ${\cal U}_{q}(SO(5))$
on suitably parametrized bases are exhibited. The implications are discussed.


title: On the Fundumental Invariant of the Hecke Algebra $H_{n}(q)$

abstract:
  The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed
class-sum of transpositions of the symmetric group $S_{n}$. Irreducible
representations of $H_{n}(q)$, for generic $q$, are shown to be completely
characterized by the corresponding eigenvalues of $C_{n}$ alone. For $S_{n}$
more and more invariants are necessary as $n$ inereases. It is pointed out that
the $q$-deformed classical quadratic Casimir of $SU(N)$ plays an analogous
role. It is indicated why and how this should be a general phenomenon
associated with $q$-deformation of classical algebras. Apart from this
remarkable conceptual aspect $C_{n}$ can provide powerful and elegant
techniques for computations. This is illustrated by using the sequence $C_{2}$,
$C_{3}, \cdots,\; C_{n}$ to compute the characters of $H_{n}(q)$.


title: The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions

abstract:
  We define a q-analog of the modified Bessel and Bessel-Macdonald functions.
As for the q-Bessel functions of Jackson there is a couple of functions of the
both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces
similarly to their classical counterpart.
  Their definition is based on the power expansions. We derive the recurrence
relations, difference equations, q-Wronskians, and an analog of asymptotic
expansions which turns out is exact in some domain if $q\neq 1$. Some relations
for the basic hypergeometric function which follow from this fact are
discussed.


title: Quantum Deformations of the Self-Duality Equation and Conformal Twistors

abstract:
  A noncommutative algebra of the complex $q$-twistors and their differentials
is considered on the basis of the quantum $GL_q (4)\times SL_q (2)$ group. Real
and pseudoreal $q$-twistors are discussed too. We consider the quantum-group
self-duality equation in the framework of the local gauge algebra of
differential forms on $q$-twistor spaces. Quantum deformations of the general
multi-instanton solutions are constructed. The corresponding noncommutative
algebras of moduli are introduced. The general $q$-instanton connection is a
function of the $q$-twistors and the $q$-moduli .


title: Quantum Homogeneous Spaces as Quantum Quotient Spaces

abstract:
  We show that certain embeddable homogeneous spaces of a quantum group that do
not correspond to a quantum subgroup still have the structure of quantum
quotient spaces. We propose a construction of quantum fibre bundles on such
spaces. The quantum plane and the general quantum two-spheres are discussed in
detail.


title: Atypical Representations of $U_{q}(sl(N))$ at Roots of Unity

abstract:
  We show how to adapt the Gelfand-Zetlin basis for describing the atypical
representation of ${\cal U}_{\displaystyle{q}}(sl(N))$ when $q$ is root of
unity. The explicit construction of atypical representation is presented in
details for $N=3$.


title: $L^{\pm}$ Operators and Quantum Enveloping Algebras

abstract:
  The relations between $L^{\pm}$ operators and the generators in the quantum
enveloping algebras are studied. The $L^{\pm}$ operators for $U_{q}A_{N}$ and
$U_{q}G_{2}$ algebras are explicitly expressed by the generators as examples.


title: Galois Relations on Knot Invariants

abstract:
  We discuss the existence of Galois relations obeyed by certain link
invariants. Some of these relations have recently been identified and exploited
within the context of CFT and Lie/Kac-Moody representation theory. These
relations should aid in computing knot invariants. They probably have an
interpretation in terms of quasitriangular (quasi-)Hopf algebras. They could
also have a topological interpretation, and may serve as a concrete model for
related ideas of Degiovanni, Drinfeld, Grothendieck, Ihara, and others.


title: $Psi$ - Vectors for Three Dimensional Models

abstract:
  In this paper we apply the method of psi-vectors to three dimensional
statistical models. This method gives the correspondence between the Bazhanov
-- Baxter model and its vertex formulation. Considering psi-vectors for the
Planar model, we obtain its self-duality.


title:
Chern-Simons theory on a lattice and a new description of 3-manifolds
  invariants

abstract:
  A new approach to the quantization of Chern-Simons theory has been developed
in recent papers of the author. It uses a "simulation" of the moduli space of
flat connections modulo the gauge group which reveals to be related to a
lattice gauge theory based on a quantum group. After a generalization of the
formalism of q-deformed gauge theory to the case of root of unity, we compute
explicitely the correlation functions associated to Wilson loops (and more
generally to graphs) on a surface with punctures, which are the interesting
quantity in the study of moduli space. We then give a new description of
Chern-Simons three manifolds invariants based on a description in terms of the
mapping class group of a surface. At last we introduce a three dimensional
lattice gauge theory based on a quantum group which is a lattice regularization
of Chern-Simons theory.


title: Vector bundles on elliptic curve and Sklyanin algebras

abstract:
  In [4] we introduce the associative algebras $Q_{n,k}(\CE,\tau)$. Recall the
definition. These algebras are labeled by discrete parameters $n,k$; $n,k$ are
integers $n>k>0$ and $n$ and $k$ have not common divisors. Then, $\CE$ is an
elliptic curve and $\tau$ is a point in $\CE$. We identify $\CE$ with
$\BC/\Gamma$, where $\Gamma$ is a lattice.


title:
Some Remarks on the Representation of the Generalized Deformed
  Oscillator Algebra

abstract:
  The classification of the representations of the generalized deformed
oscillator algebra is given together with several comments about possibility of
introducing a coproduct structure in some type of deformed oscillator algebra.


title: Algebras and Hopf Algebras in Braided Categories

abstract:
  This is an introduction for algebraists to the theory of algebras and Hopf
algebras in braided categories. Such objects generalise super-algebras and
super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided
categories C are recalled, the modules and comodules of Hopf algebras in such
categories are studied,the notion of `braided -commutative' or
`braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully
diagrammatic proof of the reconstruction theorem for a braided group aut(C) is
given. The theory has important implications for the theory of quasitriangular
Hopf algebras (quantum groups). It also includes important examples such as the
degenerate Sklyanin algebra and the quantum plane.


title: SLq(N) differential calculus from the differential calculus on GLq(N)

abstract:
  We show that the Faddeev-Pyatov SL_q(N) differential algebra is a subalgebra
of the GL_q(N) differential algebra constructed by Schupp, Watts and Zumino.


title:
Vertex operator algebras associated to modular invariant representations
  for $A_1 ^{(1)}$

abstract:
  We investigate vertex operator algebras $L(k,0)$ associated with
modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ ,
where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational
in the category $\cal O$ and find all irreducible representations in the
category of weight modules.


title:
Vertex operator algebras associated to admissible representations of
  $\hat{sl}_2$

abstract:
  The admissible modules for $\hat{sl}_2$ are studied from the point of view of
vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some
coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely
many distinguished irreducible representations for $\hat{sl}_2$, called
admissible representations. In this paper we prove that the vertex operator
algebra $L(l,0)$ associated to irreducible highest weight representation of $l$
is not rational if $l$ is not a positive integer. However if we change the
Virasoro algebra in certain way, $L(l,0)$ becomes a rational vertex operator
algebra whose irreducible representations are exactly those admissible
representations. We show that the $q$-dimensions with respect to the new
Virasoro algebra are modular functions. We aslo calculate the fusions rules.


title: TQFT and Whitehead's manifold

abstract:
  The aim of this note is to derive some invariants at infinity for open
3-manifolds in the framework of Topological Quantum Field Theories. These
invariants may be used to test if an open manifold is simply connected at
infinity as we done for Whitehead's manifold in case of the $sl_{2}({\bf
C})$-TQFT in level 4.


title: On Finite Type 3-manifold invariants IV: Comparison of Definitions

abstract:
  This paper compares the definitions of finite-type invariants due to Ohtsuki
and to Garoufalidis, showing that, residually, type 3m of the former equals
type m of the latter. It also shows that type 2m Ohtsuki invariants define knot
invariants of type 3m (first proved by Habegger).


title: On equivariant quantum cohomology

abstract:
  Using the Kontsevich's moduli space of stable maps, we define the equivariant
quantum cohomology for generalized flag varieties and make a rigorous
computation of quantum cohomology of flag varieties.


title:
Matched differential calculus on the quantum groups
  $GL_q(2,C),SL_q(2,C),C_q(2|0)$

abstract:
  We proposed the construction of the differential calculus on the quantum
group and its subgroup with the property of the natural reduction: the
differential calculus on the quantum group $GL_q(2,C)$ has to contain the
differential calculus on the quantum subgroup $SL_q(2,C)$ and quantum plane
$C_q(2|0)$ (''quantum matrjoshka''). We found, that there are two differential
calculi, associated to the left differential Maurer--Cartan 1-forms and to the
right differential 1-forms. Matched reduction take the degeneracy between the
left and right differentials. The classical limit ($q\to 1$) of the ''left''
differential calculus and of the ''right'' differential calculus is undeformed
differential calculus. The condition ${\cal D}_qG=1$ gives the differential
calculus on $SL_q(2,C)$, which contains the differential calculus on the
quantum plane $C_q(2|0)$.


title: Comment on ``Generalized $q$-oscillators and their Hopf structures''

abstract:
  In a recent paper (1994 {\sl J.\ Phys.\ A: Math.\ Gen.\ }{\bf 27} 5907), Oh
and Singh determined a Hopf structure for a generalized $q$-oscillator algebra.
We prove that under some general assumptions, the latter is, apart from some
algebras isomorphic to su$_q$(2), su$_q$(1,1), or their undeformed
counterparts, the only generalized deformed oscillator algebra that supports a
Hopf structure. We show in addition that the latter can be equipped with a
universal $\cR$-matrix, thereby making it into a quasitriangular Hopf algebra.


title:
Generalization and Deformations of Quantum Groups; Quantization of All
  Simple Lie Bi-Algebras

abstract:
  A large family of "standard" coboundary Hopf algebras is investigated. The
existence of a universal R-matrix is demonstrated for the case when the
parameters are in general position. Special values of the parameters are
characterized by the appearance of certain ideals; in this case the universal
R-matrix exists on the associated algebraic quotient. In special cases the
quotient is a "standard" quantum group; all familiar quantum groups including
twisted ones are obtained in this way. In other special cases one finds new
types of coboundary bi-algebras. A large class of first order deformations of
all these standard bi-algebras is investigated and the associated deformed
universal R-matrices have been calculated. One obtains, in particular,
universal R-matrices associated with all simple, complex Lie algebras
(classification by Belavin and Drinfeld) to first order in the deformation
parameter.


title: Finite type invariants of integral homology 3-spheres: A survey

abstract:
  This is a survey on the current status of the study of finite type invariants
of integral homology 3-spheres based on lectures given in the workshop on knot
theory at Banach International Center of Mathematics, Warsaw, July 1995. As a
new result, we show that the space of finite type invariants of integral
homology 3-spheres is a graded polynomial algebra generated by invariants
additive under the connected sum. We also discuss some open questions on this
subject.


title: Quantum Lie algebras of type A_n

abstract:
  It is shown that the quantised enveloping algebra of sl(n) contains a quantum
Lie algebra, defined by means of axioms similar to Woronowicz's., This gives
rise to Lie algebra-like generators and relations for the locally finite part
of the quantised enveloping algebra, and suggests a canonical
Poincare-Birkhoff-Witt basis.