title:
A nonlinear deformed su(2) algebra with a two-colour quasitriangular
  Hopf structure

abstract:
  Nonlinear deformations of the enveloping algebra of su(2), involving two
arbitrary functions of J_0 and generalizing the Witten algebra, were introduced
some time ago by Delbecq and Quesne. In the present paper, the problem of
endowing some of them with a Hopf algebraic structure is addressed by studying
in detail a specific example, referred to as ${\cal A}^+_q(1)$. This algebra is
shown to possess two series of (N+1)-dimensional unitary irreducible
representations, where N=0, 1, 2, .... To allow the coupling of any two such
representations, a generalization of the standard Hopf axioms is proposed by
proceeding in two steps. In the first one, a variant and extension of the
deforming functional technique is introduced: variant because a map between two
deformed algebras, su_q(2) and ${\cal A}^+_q(1)$, is considered instead of a
map between a Lie algebra and a deformed one, and extension because use is made
of a two-valued functional, whose inverse is singular. As a result, the Hopf
structure of su_q(2) is carried over to ${\cal A}^+_q(1)$, thereby endowing the
latter with a double Hopf structure. In the second step, the definition of the
coproduct, counit, antipode, and R-matrix is extended so that the double Hopf
algebra is enlarged into a new algebraic structure. The latter is referred to
as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix
is a solution of the coloured Yang-Baxter equation, where the `colour'
parameters take two discrete values associated with the two series of
finite-dimensional representations.


title: Nonlinear deformed su(2) algebras involving two deforming functions

abstract:
  The most common nonlinear deformations of the su(2) Lie algebra, introduced
by Polychronakos and Ro\v cek, involve a single arbitrary function of J_0 and
include the quantum algebra su_q(2) as a special case. In the present
contribution, less common nonlinear deformations of su(2), introduced by
Delbecq and Quesne and involving two deforming functions of J_0, are reviewed.
Such algebras include Witten's quadratic deformation of su(2) as a special
case. Contrary to the former deformations, for which the spectrum of J_0 is
linear as for su(2), the latter give rise to exponential spectra, a property
that has aroused much interest in connection with some physical problems.
Another interesting algebra of this type, denoted by ${\cal A}^+_q(1)$, has two
series of (N+1)-dimensional unitary irreducible representations, where N=0, 1,
2, .... To allow the coupling of any two such representations, a generalization
of the standard Hopf axioms is proposed. The resulting algebraic structure,
referred to as a two-colour quasitriangular Hopf algebra, is described.


title: Representation Theory of Generalized Deformed Oscillator Algebras

abstract:
  The representation theory of the generalized deformed oscillator algebras
(GDOA's) is developed. GDOA's are generated by the four operators
${1,a,a^{\dag},N}$. Their commutators and Hermiticity properties are those of
the boson oscillator algebra, except for $[a, a^{\dag}]_q = G(N)$, where
$[a,b]_q = a b - q b a$ and $G(N)$ is a Hermitian, analytic function. The
unitary irreductible representations are obtained by means of a Casimir
operator $C$ and the semi-positive operator $a^{\dag} a$. They may belong to
one out of four classes: bounded from below (BFB), bounded from above (BFA),
finite-dimentional (FD), unbounded (UB). Some examples of these different types
of unirreps are given.


title:
q-Difference Realization of U_q(sl(M|N)) and Its Application to Free
  Boson Realization of U_q(\widehat{sl}(2|1))

abstract:
  We present a q-difference realization of the quantum superalgebra
U_q(sl(M|N)), which includes Grassmann even and odd coordinates and their
derivatives. Based on this result we obtain a free boson realization of the
quantum affine superalgebra U_q(\widehat{sl}(2|1)) of an arbitrary level k.


title: On quantum group GL_{p,q}(2)

abstract:
  In the Hopf algebra GL_{p,q}(2) the determinant is central iff p=q. In this
case we put determinant to be equal to 1 to get SL_q(2). In this paper I
consider the case when p/q is a root of unity; and, consequently, a power of
the determinant is central.


title: Vertex operators for twisted quantum affine algebras

abstract:
  We construct explicitly the $q$-vertex operators (intertwining operators) for
the level one modules $V(\Lambda_i)$ of the classical quantum affine algebras
of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$,
$i=0$ for $D_4^{(3)}$, $i=0, n$ for $D_{n+1}^{(2)}$, and $i=n$ for
$A_{2n}^{(2)}$. A perfect crystal graph for $D_4^{(3)}$ is constructed as a
by-product.


title: Bosonic Realizations of $U_q(C^{(1)}_n)$

abstract:
  We construct explicitly the quantum symplectic affine algebra
$U_q(\widehat{sp}_{2n})$ using bosonic fields. The Fock space decomposes into
irreducible modules of level -1/2, quantizing the Feingold-Frenkel construction
for q=1.


title: Vertex Operators of Admissible Modules of $U_q(C^{(1)}_n)$

abstract:
  Using our recent bosonic realization of $U_q(\widehat{sp}_{2n})$, we
construct explicitly the vertex operators for the level -1/2 modules of
$U_q(\widehat{sp}_{2n})$ using bosonic fields. Our method contains a detailed
analysis of all the q-intertwining relations.


title: Cohomology of Lie superalgebras and of their generalizations

abstract:
  The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.


title: Quantization of Lie bialgebras, II

abstract:
  This paper is a continuation of "Quantization of Lie bialgebras, I"
(q-alg/9606005). We show that the quantization procedure defined in
"Quantization of Lie bialgebras, I" is given by universal acyclic formulas and
defines a functor from the category of Lie bialgebras to the category of
quantized universal enveloping algebras. We also show that this functor defines
an equivalence between the category of Lie bialgebras over k[[h]] and the
category quantized universal enveloping (QUE) algebras.


title: A $q$-analogue of the type $A$ Dunkl operator and integral kernel

abstract:
  We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a
set of degree--lowering operators on the space of polynomials in $n$ variables.
This allows the construction of raising/lowering operators with a simple action
on non-symmetric Macdonald polynomials. A bilinear series of non-symmetric
Macdonald polynomials is introduced as a $q$-analogue of the type $A$ Dunkl
integral kernel ${\cal K}_A(x;y)$. The aforementioned operators are used to
show that the function satisfies $q$-analogues of the fundamental properties of
${\cal K}_A(x;y)$.


title:
Semi-infinite $q$-wedge construction of the level 2 Fock Space of
  $U_q(\affsl{2})$

abstract:
  In this proceedings a particular example from \cite{KMPY} (q-alg/9603025) is
presented: the construction of the level 2 Fock space of $\U_q(\affsl{2})$. The
generating ideal of the wedge relations is given and the wedge space defined.
Normal ordering of wedges is defined in terms of the energy function. Normally
ordered wedges form a base of the wedge space.
  The q-deformed Fock space is defined as the space of semi-infinite wedges
with a finite number of vectors in the wedge product differing from a ground
state sequence, and endowed with a separated q-adic topology . Normally ordered
wedges form a base of the Fock space. The action of $\U_q(\affsl{2})$ on the
Fock space converges in the q-adic topology. On the Fock space the action of
bosons, which commute with the $\U_q(\affsl{2})$-action, also converges in the
q-adic topology. Hence follows the decomposition of the Fock space into
irreducible $\U_q(\affsl{2})$-modules.


title:
Quasitriangular and differential structures on bicrossproduct Hopf
  algebras

abstract:
  Let X=GM be a finite group factorisation. It is shown that the quantum double
D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is
itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where
$Y=G\times M^{op}$. This provides a class of bicrossproduct Hopf algebras which
are quasitriangular. We also construct a subgroup $Y^\theta X^\theta$
associated to every order-reversing automorphism $\theta$ of X. The
corresponding Hopf algebra $kX^\theta\cobicross k(Y^\theta)$ has the same
coalgebra as H. Using related results, we classify the first order bicovariant
differential calculi on H in terms of orbits in a certain quotient space of X.


title: On the Witten-Reshetikhin-Turaev representations of mapping class groups

abstract:
  We consider a central extension of the mapping class group of a surface with
a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs
associated to SU(2) and SO(3) induce linear representations of this group. We
show that the denominators of matrices which describe these representation over
a cyclotomic field can be restricted in many cases. In this way, we give a
proof of the known result that if the surface is a torus and there are no
colored points, the representations have finite image.


title:
q-Deformed Fock spaces and modular representations of spin symmetric
  groups

abstract:
  We use the Fock space representation of the quantum affine algebra of type
$A^{(2)}_{2n}$ to obtain a description of the global crystal basis of its basic
level 1 module. We formulate a conjecture relating this basis to decomposition
matrices of spin symmetric groups in characteristic $2n+1$.


title: Elliptic algebra $A_{q,p}(\hat{sl_2})$ in the scaling limit

abstract:
  The scaling limit $A_{\hbar,\eta}(\hat{sl_2})$ of the elliptic algebra
$A_{q,p}(\hat{sl_2})$ is investigated. The limiting algebra is defined in terms
of a continuous family of generators being Fourier harmonics of Gauss
coordinates of the $L$-operator. Ding-Frenkel isomorphism between
$L$-operator's and current descriptions of the algebra
$A_{\hbar,\eta}(\hat{sl_2})$ is established and is identified with the Riemann
problem on a strip. The representations, coalgebraic structure and intertwining
operators of the algebra are studied.


title:
Heisenberg Double Description of kappa-Poincare Algebra and
  kappa-deformed Phase Space

abstract:
  The kappa-deformed dual pair of Poincare algebra and Poincare group is
formulated in the framework of Heisenberg doubles. The covariant kappa-deformed
phase space is described in detail as a subalgebra.The realizations of proposed
algebraic scheme are considered.


title: Kappa-deformed space-time uncertainty relations

abstract:
  We discuss the kappa-deformed phase space obtained as a cross product algebra
of the deformed translations algebra and its dual configuration space. We
consider two kinds of the kappa-deformed uncertainty relations.


title:
Half-quantum groups at roots of unity, path algebras and representation
  type

abstract:
  We show that finite dimensional half-quantum groups at roots of unity
corresponding to simple Lie algebras having symmetric Cartan matrix are of wild
representation type, except for sl_2. Moreover, the underlying associative
algebra is isomorphic to an admissible quotient of the path algebra of the
Cayley graph of an abelian group. A quantum type Fourier transform enables to
describe an explicit isomorphism.


title:
Permutation-type solutions to the Yang-Baxter and other n-simplex
  equations

abstract:
  We study permutation type solutions to n-simplex equations, that is,
solutions whose R matrix can be written as a product of delta- functions
depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of
the n-simplex equation reduce to an [n(n+1)/2+1]x[n(n+1)/2+1] matrix equation
over Z_D. We have completely analyzed the 2-, 3- and 4-simplex equations in the
generic D case. The solutions show interesting patterns that seem to continue
to still higher simplex equations.


title: q-combinatorics and quantum integrability

abstract:
  The idea that a Dynkin diagram can provide one of the `spatial' variables for
an integrable difference-difference system is no news. I propose a `model'
where the only variable is of this sort.


title: A Trinomial Analogue of Bailey's Lemma and N=2 Superconformal Invariance

abstract:
  We propose and prove a trinomial version of the celebrated Bailey's lemma. As
an application we obtain new fermionic representations for characters of some
unitary as well as nonunitary models of N = 2 superconformal field theory
(SCFT). We also establish interesting relations between N = 1 and N = 2 models
of SCFT with central charges $(3/2)( 1 -{2(2 - 4\nu)^2}/{2(4\nu)})$ and $3(1 -
2/{4\nu})$. A number of new mock theta function identities are derived.


title: The Fundamental Theorem of Vassiliev Invariants

abstract:
  The "fundamental theorem of Vassiliev invariants" says that every weight
system can be integrated to a knot invariant. We discuss four different
approaches to the proof of this theorem: a topological/combinatorial approach
following M. Hutchings, a geometrical approach following Kontsevich, an
algebraic approach following Drinfel'd's theory of associators, and a physical
approach coming from the Chern-Simons quantum field theory. Each of these
approaches is unsatisfactory in one way or another, and hence we argue that we
still don't really understand the fundamental theorem of Vassiliev invariants.


title: Multibraces on the Hochschild complex

abstract:
  We generalize the coupled braces {x}{y} of Gerstenhaber and {x}{y,...,z} of
Getzler depicting compositions of multilinear maps in the Hochschild complex
C(A)=Hom(TA;A) of a graded vector space A to expressions of the form
{x,...,y}...{z,...,w} on the extended space Hom(TA;TA), and clarify many of the
existing sign conventions that show up in the algebra of mathematical physics
(namely in associative and Lie algebras, Batalin-Vilkovisky algebras, homotopy
associative and homotopy Lie algebras). As a result, we introduce a new variant
of the master identity for homotopy Lie algebras. We also comment on the
bialgebra cohomology differential of Gerstenhaber and Schack, and define
multilinear higher order differential operators with respect to multilinear
maps using the new language. The continuation of this work will be on the
various homotopy structures on a topological vertex operator algebra, as
introduced by Kimura, Voronov, and Zuckerman.


title:
Matrix elements of vertex operators of deformed W-algebra and Harish
  Chandra Solutions to Macdonald's difference equations

abstract:
  In this paper we prove that certain matrix elements of vertex operators of
deformed W-algebra satisfy Macdonald difference equations and form n!
-dimensional space of solutions. These solutions are the analogues of Harish
Chandra solutions with prescribed asymptotic behavior. We obtain formulas for
analytic continuation as a consequence of braiding properties of vertex
operators of deformed W-algebra.


title: Deformation Theory and the Batalin-Vilkovisky Master Equation

abstract:
  The Batalin-Vilkovisky master equations, both classical and quantum, are
precisely the integrability equations for deformations of algebras and
differential algebras respectively. This is not a coincidence; the
Batalin-Vilkovisky approach is here translated into the language of deformation
theory.


title:
Representations of the Lie algebra gl(infinity)^ and ``reciprocity
  formula'' for Clebsch-Gordan coefficients

abstract:
  We compare two approaches to the calculation of irreducible characters of the
Lie algebra gl(infinity)^ with negative integral central charge. As a
consequence, we obtain a "reciprocity formula" for Clebsch-Gordan coefficients,
in the limit when the central charge tends to infinity.


title: Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes

abstract:
  We give a definition of weak n-categories based on the theory of operads. We
work with operads having an arbitrary set S of types, or `S-operads', and given
such an operad O, we denote its set of operations by elt(O). Then for any
S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O.
Letting I be the initial operad with a one-element set of types, and defining
I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional
opetopes'. Opetopes form a category, and presheaves on this category are called
`opetopic sets'. A weak n-category is defined as an opetopic set with certain
properties, in a manner reminiscent of Street's simplicial approach to weak
omega-categories. Similarly, starting from an arbitrary operad O instead of I,
we define `n-coherent O-algebras', which are n times categorified analogs of
algebras of O. Examples include `monoidal n-categories', `stable n-categories',
`virtual n-functors' and `representable n-prestacks'. We also describe how
n-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra.


title: Homological algebra of homotopy algebras

abstract:
  We define closed model category structures on different categories connected
to the world of operad algebras over the category C(k) of (unbounded) complexes
of k-modules: on the category of operads, on the category of algebras over a
fixed operad, on the category of modules over a fixed operad algebra. In
Sections 2 - 6 we define the necessary structures and provide some standard
comparison results. In Section 7 we define cotangent complex. In Section 8 we
define a canonical structure of homotopy Lie algebra on the tangent complex.


title:
Drinfeld-Sokolov reduction for difference operators and deformations of
  W-algebras. II. General Semisimple Case

abstract:
  The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov
reduction procedure to q-difference operators associated with arbitrary
semisimple Lie algebras. This leads to a new elliptic deformation of the Lie
bialgebra structure on the associated loop algebra. The related classical
r-matrix is explicitly described in terms of the Coxeter transformation. We
also present a cross-section theorem for q-gauge transformations which
generalizes a theorem due to R.Steinberg.


title: Idempotents of Hecke algebras of type A

abstract:
  We use a skein-theoretic version of the Hecke algebras of type A to present
three-dimensional diagrammatic views of Gyoja's idempotent elements, based
closely on the corresponding Young diagram. In this context we give
straightforward calculations for the eigenvalues of two natural central
elements in the Hecke algebras, namely the full curl and the sum of the Murphy
operators. We discuss their calculation also in terms of the framing factor
associated to the appropriate irreducible representation of the quantum group
SU(N,q).


title: An example of a braided category of C*- algebras

abstract:
  A braided category of C*-algebras is constructed. Its objects are C*-algebras
endowed with an action of the group R, its morphisms are C*-algebras morphisms
intertwining the action of R, the crossed product of its two objects
essentially depends on the action of R on considered C*-algebras. Crossed
products of any object with the algebra of all continuous vanishing at infinity
functions on R and algebras of all continuous functions on one- and
twodimensional tori are discussed.


title:
On $q$-Clebsch Gordan Rules and the Spinon Character Formulas for Affine
  $C_2^{(1)}$ Algebra

abstract:
  A $q$-analog of the Clebsch Gordan rules for the tensor products of the
fundamental representations of Yangian is introduced. Its relation to the
crystal base theory and application to the spinon character formulas are
discussed in case of $C_2^{(1)}$ explicitly.


title: On the algebraic structure of differential calculus on quantum groups

abstract:
  Intrinsic Hopf algebra structure of the Woronowicz differential complex is
shown to generate quite naturally a bicovariant algebra of four basic objects
within a differential calculus on quantum groups -- coordinate functions,
differential 1-forms, Lie derivatives, and inner derivations -- as the
cross-product algebra of two mutually dual graded Hopf algebras. This
construction, properly taking into account Hopf-algebraic properties of
Woronowicz's bicovariant calculus, provides a direct proof of the Cartan
identity and of many other useful relations. A detailed comparison with other
approaches is also given.


title: A note on the trinomial analogue of Bailey's lemma

abstract:
  Recently, Andrews and Berkovich introduced a trinomial version of Bailey's
lemma. In this note we show that each ordinary Bailey pair gives rise to a
trinomial Bailey pair. This largely widens the applicability of the trinomial
Bailey lemma and proves some of the identities proposed by Andrews and
Berkovich.


title: Difference Macdonald-Mehta Conjecture

abstract:
  In the paper we formulate and verify a difference counterpart of the
Macdonald-Mehta conjecture and its generalization for the Macdonald
polynomials. Namely, we determine the Fourier transforms of the polynomials
multiplied by the Gaussian, which is closely connected with the new difference
Harish-Chandra theory.


title:
Quasiclassical Limit in q-Deformed Systems, Noncommutativity and the
  q-Path Integral

abstract:
  Different analogs of quasiclassical limit for a q-oscillator which result in
different (commutative and non-commutative) algebras of ``classical''
observables are derived. In particular, this gives the q-deformed Poisson
brackets in terms of variables on the quantum planes. We consider the
Hamiltonian made of special combination of operators (the analog of even
operators in Grassmann algebra) and discuss q-path integrals constructed with
the help of contracted ``classical'' algebras.


title: Toroidal actions on level 1 modules of U_q(\hat{sl_n})

abstract:
  Recently Varagnolo and Vasserot established that the q-deformed Fock spaces
due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum
toroidal algebra $U_q(sl_n,tor)$ (n > 2) with the level (0,1). In the present
article we propose a more detailed proof of this fact then the one given by
Varagnolo and Vasserot. The proof is based on certain non-trivial properties of
Cherednik's commuting difference operators. The quantum toroidal action on the
Fock space depends on a certain parameter. We find that with a specific choice
of this parameter the action on the Fock spaces gives rise to the toroidal
action on irreducible level-1 highest weight modules of the affine quantum
algebra $U_q(\hat{sl}_n)$. Similarly, by a specific choice of the parameter,
the level (1,0) vertex representation of the quantum toroidal algebra gives
rise to a $U_q(sl_n,tor)$-module structure on irreducible level-1 highest
weight $U_q(\hat{sl}_n)$-modules.


title: Non-commutative Euclidean and Minkowski Structure

abstract:
  A noncommutative *-algebra that generalizes the canonical commutation
relations and that is covariant under the quantum groups SOq(3) or SOq(1,3) is
introduced. The generating elements of this algebra are hermitean and can be
identified with coordinates, momenta and angular momenta. In addition a unitary
scaling operator is part of the algebra.


title: Twisted h-spacetimes and invariant equations

abstract:
  We analyze the h-deformations of the Lorentz group and their associated
spacetimes. We prove that they have a twisted character and give explicitly the
twisting matrices. After studying the representations of one of the deformed
spacetime algebras, we discuss the Klein-Gordon operator. It is found that the
h-deformed d'Alembertian has plane wave solutions of the same form as the
standard ones. We also give explicit expressions for the h-gamma matrices
defining the associated Dirac equations.


title:
Twisted representations of vertex operator algebras and associative
  algebras

abstract:
  Let V be a vertex operator algebra and g an automorphism of order T. We
construct a sequence of associative algebras A_{g,n}(V) with n\in\frac{1}{T}\Z
nonnegative such that A_{g,n}(V) is a quotient of A_{g,n+1/T}(V) and a pair of
functors between the category of A_{g,n}(V)-modules which are not
A_{g,n-1/T}(V)-modules and the category of admissible V-modules. These functors
exhibit a bijection between the simple modules in each category. We also show
that V is g-rational if and only if all A_{g,n}(V) are finite-dimensional
semisimple algebras.


title:
Twisting 2-cocycles for the construction of new non-standard quantum
  groups

abstract:
  We introduce a new class of 2-cocycles defined explicitly on the generators
of certain multiparameter standard quantum groups. These allow us, through the
process of twisting the familiar standard quantum groups, to generate new as
well as previously known examples of non-standard quantum groups. In particular
we are able to construct generalisations of both the Cremmer-Gervais
deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and
Galindo in an explicit and straightforward manner.


title: Algebra of screening operators for the deformed $W_n$ algebra

abstract:
  We construct a family of intertwining operators (screening operators) between
various Fock space modules over the deformed $W_n$ algebra. They are given as
integrals involving a product of screening currents and elliptic theta
functions. We derive a set of quadratic relations among the screening
operators, and use them to construct a Felder-type complex in the case of the
deformed $W_3$ algebra.


title: Shadow of noncommutativity

abstract:
  We analyse the structure of the $\kappa=0$ limit of a family of algebras
$A_\kappa$ describing noncommutative versions of space-time, with $\kappa$ a
parameter of noncommutativity. Assuming the Poincar\'e covariance of the
$\kappa=0$ limit, we show that, besides the algebra of functions on Minkowski
space, $A_0$ must contain a nontrivial extra factor $A^I_0$ which is Lorentz
covariant and which does not commute with the functions whenever it is not
commutative. We give a general description of the possibilities and analyse
some representative examples.


title: Contractions on the Classical Double

abstract:
  Lie algebra contractions on the classical Drinfel'd Double of a given Lie
bialgebra are introduced and compared to the usual Lie bialgebra contraction
theory. The connection between both approaches turns out to be intimately
linked to duality problems. The non-relativistic (Galilean) limit of a (1+1)
Poincar\'e Double is used to illustrate the contraction process. Finally, it is
shown that, in a certain sense, the classical limit in a quantum algebra can be
thought as a certain contraction on the corresponding Double.


title:
Cartan-Calculus and its Generalizations via a Path-Integral Approach to
  Classical Mechanics

abstract:
  In this paper we review the recently proposed path-integral counterpart of
the Koopman-von Neumann operatorial approach to classical Hamiltonian
mechanics. We identify in particular the geometrical variables entering this
formulation and show that they are essentially a basis of the cotangent bundle
to the tangent bundle to phase-space. In this space we introduce an extended
Poisson brackets structure which allows us to re-do all the usual Cartan
calculus on symplectic manifolds via these brackets. We also briefly sketch how
the Schouten-Nijenhuis, the Fr\"olicher- Nijenhuis and the Nijenhuis-Richardson
brackets look in our formalism.


title: Twisting of monoidal structures

abstract:
  This article is devoted to the investigation of the deformation (twisting) of
monoidal structures, such as the associativity constraint of the monoidal
category and the monoidal structure of monoidal functor. The sets of twistings
have a (non-abelian) c ohomological nature. Using this fact the maps from the
sets of twistings to some cohomology groups (Hochschild cohomology of K-theory)
are constructed. The examples of monoidal categories of bimodules over some
algebra, modules and comodules over bialgeb ra are examined. We specially
concentrate on the case of free tensor category.


title:
A generalization of Scheunert's Theorem on cocycle twisting of color Lie
  algebras

abstract:
  A classical theorem of Scheunert on $G$-color Lie algebras, asserts in the
case of finitely generated abelian groups, one can twist the algebra structure
and the commutation bicharacter on $G$ by a 2-cocycle twist to a super-Lie $G$
graded, algebra. In this paper we show that this can be done for an arbitrary
group.


title: Row-reducing the quantum determinant and Dieudonne determinant

abstract:
  We prove that row reducing a quantum matrix yields another quantum matrix for
the same parameter q. This means that the elements of the new matrix satisfy
the same relations as those of the original quantum matrix ring M_q(n). As a
corollary, we can prove that the image of the quantum determinant in the
abelianization of the total ring of quotients of M_q(n) is equal to the
Dieudonne determinant of the quantum matrix. A similar result is proved for the
multiparameter quantum determinant.


title: Braided-Lie bialgebras

abstract:
  We introduce braided Lie bialgebras as the infinitesimal version of braided
groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie
cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie
biproduct, bosonisation and double-bosonisation relating braided Lie bialgebras
to usual Lie bialgebras. Among the results, the kernel of any split projection
of Lie bialgebras is a braided-Lie bialgebra. The Kirillov-Kostant Lie
cobracket provides a natural braided-Lie bialgebra on any complex simple Lie
algebra $g$, as the transmutation of the Drinfeld-Sklyanin Lie cobracket. Other
nontrivial braided-Lie bialgebras are associated to the inductive construction
of simple Lie bialgebras along the $C$ and exceptional series.


title: On q-Analogues of Bounded Symmetric Domains and Dolbeault Complexes

abstract:
  A very well known result by Harish-Chandra claims that any Hermitian
symmetric space of non-compact type admits a canonical embedding into a complex
vector space $V$. The image of this embedding is a bounded symmetric domain in
$V$. This work provides a construction of q-analogues of a polynomial algebra
on $V$ and the differential algebra of exterior forms on $V$. A way of
producing a q-analogue of the bounded function algebra in a bounded symmetric
domain is described. All the constructions are illustrated by detailed
calculations in the case of the simplest Hermitian symmetric space
$SU(1,1)/U(1)$. The development of these ideas can be found in math.QA/9803110
and math.QA/9809038 .


title:
Generalized Hermite Polynomials and the Heat Equation for Dunkl
  Operators

abstract:
  Based on the theory of Dunkl operators, this paper presents a general concept
of multivariable Hermite polynomials and Hermite functions which are associated
with finite reflection groups on $\b R^N$. The definition and properties of
these generalized Hermite systems extend naturally those of their classical
counterparts; partial derivatives and the usual exponential kernel are here
replaced by Dunkl operators and the generalized exponential kernel K of the
Dunkl transform. In case of the symmetric group $S_N$, our setting includes the
polynomial eigenfunctions of certain Calogero-Sutherland type operators. The
second part of this paper is devoted to the heat equation associated with
Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem
is governed by a positive one-parameter semigroup; this is assured by a maximum
principle for the generalized Laplacian. The explicit solution to the Cauchy
problem involves again the kernel K, which is, on the way, proven to be
nonnegative for real arguments.


title: Canonical basis and homology of local systems

abstract:
  Using the isomorphism between highest weight U_q(sl_2)-modules and homologies
of certain local systems on the configuration spaces, constructed by Varchenko,
we give a geometric construction of the dual of the Lusztig's canonical basis
in a tensor product of irreducible finite-dimensional U_q(sl_2)-modules.


title:
Towards constructing one-particle representations of the deformed
  Poincar\'e algebra

abstract:
  We give a method for obtaining states of massive particle representations of
the two-parameter deformation of the Poincar\'e algebra proposed in
q-alg/9601010, q-alg/9505030 and q-alg/9501026. We discuss four procedures to
generate eigenstates of a complete set of commuting operators starting from the
rest state. One result of this work is the fact that upon deforming to the
quantum Poincar\'e algebra the rest state is split into an infinite number of
states. Another result is that the energy spectrum of these states is discrete.
Some curious residual degeneracy remains: there are states constructed by
applying different operators to the rest state which nevertheless are
indistinguishable by eigenvalues of all the observables in the algebra.


title:
Bargmann representation for some deformed harmonic oscillators with
  non-Fock representation

abstract:
  We prove that Bargmann representations exist for some deformed harmonic
oscillators that admit non-Fock representations. In specific cases, we
explicitly obtain the resolution of the identity in terms of a true integral on
the complex plane. We prove on explicit examples that Bargmann representations
cannot always be found, particularly when the coherent states do not exist in
the whole complex plane.


title: Kazhdan-Lusztig tensoring and Harish-Chandra categories

abstract:
  We use the Kazhdan-Lusztig tensoring to define affine translation functors,
describe annihilating ideals of highest weight modules over an affine Lie
algebra in terms of the corresponding VOA, and to sketch a functorial approach
to ``affine Harish-Chandra bimodules''.


title:
Representations and Clebsch-Gordan coefficients for the Jordanian
  quantum algebra U_h(sl(2))

abstract:
  Representation theory for the Jordanian quantum algebra $U=U_h(sl(2))$ is
developed. Closed form expressions are given for the action of the generators
of U on the basis vectors of finite dimensional irreducible representations. It
is shown how representation theory of U has a close connection to combinatorial
identities involving summation formulas. A general formula is obtained for the
Clebsch-Gordan coefficients $C^{j_1,j_2,j}_{n_1,n_2,m}(h)$ of U. These
Clebsch-Gordan coefficients are shown to coincide with those of su(2) for
$n_1+n_2 \leq m$, but for $n_1+n_2 > m$ they are in general a nonzero monomial
in $h^{n_1+n_2-m}$.


title:
A construction of Hopf algebra cocycles for the double Yangian
  $DY(SL_{2})$

abstract:
  We construct a Hopf algebra cocycle in the Yangian double $DY(SL_{2})$,
conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the
twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in
earlier work by V. Rubtsov and the first author, using the decomposition of the
algebra in its negative and non-negative modes subalgebras.


title: Factorisation of Macdonald polynomials

abstract:
  We discuss the problem of factorisation of the symmetric Macdonald
polynomials and present the obtained results for the cases of 2 and 3
variables.


title: The Dirac operator and gamma matrices for quantum Minkowski spaces

abstract:
  Gamma matrices for quantum Minkowski spaces are found. The invariance of the
corresponding Dirac operator is proven. We introduce momenta for spin 1/2
particles and get (in certain cases) formal solutions of the Dirac equation.


title: Free coherent states and distributions on p-adic numbers

abstract:
  Free coherent states for a system with two degrees of freedom is defined. An
isomorphism of the space of distributions on 2-adic disc and the space of free
coherent states is constructed.


title: Modular invariance of trace functions in orbifold theory

abstract:
  The goal of the present paper is to provide a mathematically rigorous
foundation to certain aspects of rational orbifold conformal field theory, in
other words the theory of rational vertex operator algebras and their
automorphisms. Under a certain finiteness condition on a rational vertex
operator algebra V which holds in all known examples, we determine the precise
numbers of g-twisted sectors for any automorphism g of V of finite order. We
prove that the trace functions and correlations functions associated with such
twisted sectors are holomorphic functions in the upper half-plane and, under
suitable conditions, afford a representations of the modular group of the type
prescribed in string theory. We establish the rationality of conformal weights
and central charge. In addition to conformal field theory itself, where our
conclusions are required on physical grounds, there are applications to the
generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant
elliptic cohomology.


title: Traces of creation-annihilation operators and Fredholm's formulas

abstract:
  We prove the formula for the traces of certain class of operators in bosonic
and fermionic Fock spaces. Vertex operators belong to this class. Traces of
vertex operators can be used for calculation of correlation functions and
formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are
interested in this problem. Also we show that Fredholm's minor and determinant
can be expressed by such traces. We obtain a short proof of the Fredholm's
formula for the solution of an integral equation. Analogue of Fredholm's
formula in terms of permanents is obtained.


title: Elliptic quantum groups $E_{\tau,\eta}(sl_2)$ and quasi-Hopf algebras

abstract:
  We construct an algebra morphism from the elliptic quantum group
$E_{\tau,\eta}(sl_2)$ to a certain elliptic version of the ``quantum groups in
higher genus'' studied by V. Rubtsov and the first author. This provides an
embedding of $E_{\tau,\eta}(sl_2)$ in an algebra ``with central extension''. In
particular we construct $L^{\pm}$-operators obeying a dynamical version of the
Reshetikhin--Semenov-Tian-Shansky relations. To do that, we construct the
factorization of a certain twist of the latter algebra, that automatically
satisfies the ``twisted cocycle condition'' of O. Babelon, D. Bernard and E.
Billey, and therefore provides a solution of the dynamical Yang-Baxter
equation.


title: Nambu mechanics, $n$-ary operations and their quantization

abstract:
  We start with an overview of the "generalized Hamiltonian dynamics"
introduced in 1973 by Y. Nambu, its motivations, mathematical background and
subsequent developments -- all of it on the classical level. This includes the
notion (not present in Nambu's work) of a generalization of the Jacobi identity
called Fundamental Identity. We then briefly describe the difficulties
encountered in the quantization of such $n$-ary structures, explain their
reason and present the recently obtained solution combining deformation
quantization with a "second quantization" type of approach on ${\Bbb R}^n$. The
solution is called "Zariski quantization" because it is based on the
factorization of (real) polynomials into irreducibles. Since we want to
quantize composition laws of the determinant (Jacobian) type and need a Leibniz
rule, we need to take care also of derivatives and this requires going one step
further (Taylor developments of polynomials over polynomials). We also discuss
a (closer to the root, "first quantized") approach in various circumstances,
especially in the case of covariant star products (exemplified by the case of
su(2)). Finally we address the question of equivalence and triviality of such
deformation quantizations of a new type (the deformations of algebras are more
general than those considered by Gerstenhaber).


title:
Comments on Drinfeld Realization of Quantum Affine Superalgebra
  $U_q[gl(m|n)^{(1)}]$ and its Hopf Algebra Structure

abstract:
  By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to
supersymmetric cases, we obtain Drinfeld current realization for quantum affine
superalgebra $U_q[gl(m|n)^{(1)}]$. We find a simple coproduct for the quantum
current generators and establish the Hopf algebra structure of this super
current algebra.


title: Spinon decomposition and Yangian structure of (sl_n)^ modules

abstract:
  We review several fermionic-type character formulae for the characters of the
integrable highest weight modules of sl(n)^ at level l=1, and explain how they
arise from a spinon basis for these modules. We also review how the Yangian
Y(sl(n)) acts on the integrable (sl(n)^)_1 modules and we decompose the
characters with respect to this Y(sl(n)) action.


title:
Drinfel'd Realization of Quantum Affine Superalgebra
  $U_q\hat{(gl(1|1))}$

abstract:
  We obtain Drinfel'd's realization of quantum affine superalgebra
$U_q\hat{(gl(1|1))}$ based on the super version of RS construction method and
Gauss decomposition.


title: Quantum and Classical Integrable Systems

abstract:
  The key concept discussed in these lectures is the relation between the
Hamiltonians of a quantum integrable system and the Casimir elements in the
underlying hidden symmetry algebra. (In typical applications the latter is
either the universal enveloping algebra of an affine Lie algebra, or its
q-deformation.) A similar relation also holds in the classical case. We discuss
different guises of this very important relation and its implication for the
description of the spectrum and the eigenfunctions of the quantum system.
Parallels between the classical and the quantum cases are thoroughly discussed.


title: The binomial formula for nonsymmetric Macdonald polynomials

abstract:
  Inhomogeneous analogues of symmetric and nonsymmetric Macdonald polynomials
were introduced by F. Knop and the author. In the symmetric case A. Okounkov
has recently proved a beautiful expansion formula which can be viewed as a
multivariable generalization of the q-binomial theorem.
  It is becoming clear that many interesting results in the symmetric setting
can also be established in the nonsymmetric setting. In this paper, we prove
the nonsymmetric analogue the binomial formula, and also discuss the limiting
case of Jack polynomials.


title: Wheels, Wheeling, and the Kontsevich Integral of the Unknot

abstract:
  We conjecture an exact formula for the Kontsevich integral of the unknot, and
also conjecture a formula (also conjectured independently by Deligne) for the
relation between the two natural products on the space of Chinese characters.
The two formulas use the related notions of "Wheels" and "Wheeling". We prove
these formulas "on the level of Lie algebras" using standard techniques from
the theory of Vassiliev invariants and the theory of Lie algebras.


title: q-Deformed Schr\"odinger Equation

abstract:
  We found hermitian realizations of the position vector $\vec{r}$, angular
momentum $\vec{\Lambda}$ and linear momentum $\vec{p}$ behaving like vectors
with respect to the $SU_q(2)$ algebra, generated by $L_0$ and $L_\pm$. They are
used to write the $q$ deformed Schr\"odinger equation, whose solution for
Coulomb and oscillator potential are briefly discussed.


title: Super Yangian Double and its Central Extension

abstract:
  We introduce super Yangian double $DY_\hbar[gl(m|n)]$ and its central
extension $\widehat{DY_\hbar[gl(m|n)]}$. We give their defining relations in
terms of current generators and obtain Drinfeld comultiplication.


title: Finite-dimensional Representations of Quantum Affine Algebras

abstract:
  We present a conjecture on the irreducibility of the tensor products of
fundamental representations of quantized affine algebras. This conjecture
implies in particular that the irreducibility of the tensor products of
fundamental representations is completely described by the poles of the
R-matrices. The conjecture is proved in certain cases.


title: Certain Topics on the Lie Algebra gl(lambda) Representation Theory

abstract:
  The Lie algebra gl(lambda) dependent on the complex parameter lambda is a
continuous version of the Lie algebra gl(inf) of infinite matrices with only
finite number of nonzero entries. The gl(lambda) was first introduced by
B.L.Feigin in [1] in connection with the Lie algebra cohomologies of the
differential operators on the complex line. The paper is devoted to the
representation theory of the gl(lambda). The Shapovalov's form determinant is
calculated; it depends polynomially on lambda and the parameters of the
representation. It is natural to include the Hamiltonian Lie algebra of
functions on the hyperboloid to the range of gl(lambda), relating it to the
value lambda = infinity. Thus, every family of representations of gl(lambda)
depending holomorphically on lambda is associated with a holomorphic vector
bundle on the Riemann sphere. The determinant of the Shapovalov's form thus
determines the corresponding line bundle. The Chern class of this bundle is
calculated in two different ways (for every level on the representation). The
comparison of the two formulas yields new combinatorial identities with power
series.
  The results of Chapter 1 about the Lie algebra gl(inf) may have some
independent interest.


title:
Two-parameter Quantum Affine Superalgebra $U_{p,q}\widehat{(gl(1|1))}$
  and Its Drinfeld's Realization

abstract:
  Using super RS construction method and Gauss decomposition, we obtain
Drinfeld's currents realization of two-parameter quantum affine superalgebra
$U_{p,q}\widehat{(gl(1|1))}$ and get co-product structure for these currents.


title: Super Yangian Double DY(gl(1|1)) With Center Extension

abstract:
  We use the super version of RS construction method and Gauss decomposition to
obtain Drinfeld's currents realization of super Yangian double $DY(gl(1|1))$
with center extension


title: Green Function on the q-Symmetric Space SU_q(2)/U(1)

abstract:
  Following the introduction of the invariant distance on the non-commutative
C-algebra of the quantum group SU_q(2), the Green function and the Kernel on
the q-homogeneous space M=SU(2)_q/U(1) are derived. A path integration is
formulated. Green function for the free massive scalar field on the
non-commutative Einstein space R^1xM is presented.


title: 2-Tangles

abstract:
  Just as links may be algebraically described as certain morphisms in the
category of tangles, compact surfaces smoothly embedded in R^4 may be described
as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. In
this announcement we give a purely algebraic characterization of the 2-category
of unframed unoriented 2-tangles in 4 dimensions as the `free semistrict
braided monoidal 2-category with duals on one unframed self-dual object'. A
forthcoming paper will contain a proof of this result using the movie moves of
Carter, Rieger and Saito. We comment on how one might use this result to
construct invariants of 2-tangles.


title:
Non Commutative Differential Geometry, and the Matrix Representations of
  Generalised Algebras

abstract:
  The underlying algebra for a noncommutative geometry is taken to be a matrix
algebra, and the set of derivatives the adjoint of a subset of traceless
matrices. This is sufficient to calculate the dual 1-forms, and show that the
space of 1-forms is a free module over the algebra of matrices. The concept of
a generalised algebra is defined and it is shown that this is required in order
for the space of 2-forms to exist. The exterior derivative is generalised for
higher order forms and these are also shown to be free modules over the matrix
algebra. Examples of mappings that preserve the differential structure are
given. Also given are four examples of matrix generalised algebras, and the
corresponding noncommutative geometries, including the cases where the
generalised algebra corresponds to a representation of a Lie algebra or a
$q$-deformed algebra.


title: Path integrals and low-dimensional topology

abstract:
  A path-integral approach to non-perturbative topological invariants of knots,
links and manifolds of dimension three and four using topological quantum field
theory of Schwarz (Chern-Simons) type is presented.


title: Differential Calculus in Braided Abelian Categories

abstract:
  Braided non-commutative differential geometry is studied. In particular we
investigate the theory of (bicovariant) differential calculi in braided abelian
categories. Previous results on crossed modules and Hopf bimodules in braided
categories are used to construct higher order bicovariant differential calculi
over braided Hopf algebras out of first order ones. These graded objects are
shown to be braided differential Hopf algebras with universal bialgebra
properties. The article especially extends Woronowicz's results on
(bicovariant) differential calculi to the braided non-commutative case.


title: The boundary of Young graph with Jack edge multiplicities

abstract:
  Consider the lattice of all Young diagrams ordered by inclusion, and denote
by Y its Hasse graph. Using the Pieri formula for Jack symmetric polynomials,
we endow the edges of the graph Y with formal multiplicities depending on a
real parameter $\theta$. The multiplicities determine a potential theory on the
graph Y. Our main result identifies the corresponding Martin boundary with an
infinite-dimensional simplex, the ``geometric boundary'' of the Young graph Y,
and provides a canonical integral representation for non-negative harmonic
functions.
  For three particular values of the parameter, the theorem specializes to
known results: the Thoma theorem describing characters of the infinite
symmetric group, the Kingman's classification of partition structures, and the
description of spherical functions of the infinite hyperoctahedral Gelfand
pair.


title:
A Natural Basis of States for the Noncommutative Sphere and its Moyal
  bracket

abstract:
  An infinite dimensional algebra which is a non-decomposable reducible
representation of $su(2)$ is given. This algebra is defined with respect to two
real parameters. If one of these parameters is zero the algebra is the
commutative algebra of functions on the sphere, otherwise it is a
noncommutative analogue. This is an extension of the algebra normally refered
to as the (Berezin) quantum sphere or ``fuzzy'' sphere. A natural indefinite
``inner'' product and a basis of the algebra orthogonal with respect to it are
given. The basis elements are homogenious polynomials, eigenvectors of a
Laplacian, and related to the Hahn polynomials. It is shown that these elements
tend to the spherical harmonics for the sphere. A Moyal bracket is constructed
and shown to be the standard Moyal bracket for the sphere.


title: Quommutator deformations of spl(N,1)

abstract:
  We analyse the Witten-Woronowicz's type deformations of the Lie superalgebras
spl(N,1) and obtain deformations parametrized by N(N-1)/2 independent
parameters for N>2 and by three parameters for N=2. For some of these algebras,
finite dimensional representations can be constructed in terms of finite
difference operators, providing operators that are relevant for the
classification of quasi exactly solvable, finite difference equations.


title:
Geometry and classification of solutions of the Classical Dynamical
  Yang-Baxter Equation

abstract:
  The classical Yang-Baxter equation (CYBE) is an algebraic equation central in
the theory of integrable systems. Its solutions were classified by Belavin and
Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric
interpretation of CDYB was given by Drinfeld and gave rise to the theory of
Poisson-Lie groups. The classical dynamical Yang-Baxter equation (CDYBE) is an
important differential equation analagous to CYBE and introduced by Felder as
the consistency condition for the Knizhnik-Zamolodchikov-Bernard equations for
correlation functions in conformal field theory on tori. Quantization of CDYBE
allowed Felder to introduce an interesting elliptic analog of quantum groups.
It becomes clear that numerous important notions and results connected with
CYBE have dynamical analogs. In this paper we classify solutions to CDYBE and
give geometric interpretation to CDYBE. The classification and interpretation
are remarkably analogous to the Belavin-Drinfeld picture.


title: Dual quasitriangular structures related to the Temperley-Lieb algebra

abstract:
  We consider nonquasiclassical solutions to the quantum Yang-Baxter equation
and the corresponding quantum cogroups $\Fun(SL(S))$ constructed earlier by one
of the authors . We give a criterion of the existence of a dual quasitriangular
structure in the algebra $\Fun(SL(S))$ and describe a large class of such
objects related to the Temperley-Lieb algebra satisfying this criterion. We
show also that this dual quasitriangular structure is in some sense
nondegenerate.


title: Two types of Poisson pencils and related quantum objects

abstract:
  Two types of Poisson pencils connected to classical R-matrices and their
quantum counterparts are considered. A representation theory of the quantum
algebras related to some symmetric orbits in $sl(n)^*$ is constructed. A
twisted version of quantum mechanics is discussed.


title: Classical Limit of the Scaled Elliptic Algebra $A_{\hbar,\eta}(sl_2)$

abstract:
  The classical limit of the scaled elliptic algebra $A_{\hbar,\eta}(sl_2)$ is
investigated. The limiting Lie algebra is described in two equivalent ways: as
a central extension of the algebra of generalized automorphic $sl_2$ valued
functions on a strip and as an extended algebra of decreasing automorphic
$sl_2$ valued functions on the real line. A bialgebra structure and an
infinite-dimensional representation in the Fock space are studied. The
classical limit of elliptic algebra $A_{q,p}(sl_2)$ is also briefly presented.


title:
Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and
  Elliptic Quantum Groups

abstract:
  The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the quantum group $U_q(sl_2)$ is a system of linear difference
equations with values in a tensor product of $U_q(sl_2)$ Verma modules. We
solve the equation in terms of multidimensional $q$-hypergeometric functions
and define a natural isomorphism between the space of solutions and the tensor
product of the corresponding evaluation Verma modules over the elliptic quantum
group $E_{\rho,\gamma}(sl_2)$, where parameters $\rho$ and $\gamma$ are related
to the parameter $q$ of the quantum group $U_q(sl_2)$ and the step $p$ of the
qKZ equation via $p=e^{2\pii\rho}$ and $q=e^{-2\pii\gamma}$.
  We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the dynamical elliptic R-matrices. This description of the transition
functions gives a connection between representation theories of the quantum
loop algebra $U_q(\widetilde{gl}_2$ and the elliptic quantum group
$E_{\rho,\gamma}(sl_2)$ and is analogous to the Kohno-Drinfeld theorem on the
monodromy group of the differential Knizhnik-Zamolodchikov equation.
  In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.


title:
Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody
  Algebras

abstract:
  Let B(\infty) be the crystal corresponding to the nilpotent part of a
quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as
the set of integer solutions of a system of linear inequalities. As an
application, we treat in a unified manner all Kac-Moody algebras of rank 2
(sharpening the result by Kashiwara), as well as the algebras of types A_n and
A_{n-1}^{(1)}.


title: The algebra A_{\hbar,\eta}(\hat{g}) and Infinite Hopf family of algebras

abstract:
  New deformed affine algebras A_{\hbar,\eta}(\hat{g}) are defined for any
simply-laced classical Lie algebra g, which are generalizations of the algebra
A_{\hbar,\eta}(\hat{sl_2}) recently proposed by Khoroshkin, Lebedev and
Pakuliak (KLP). Unlike the work of KLP, we associate to the new algebras the
structure of an infinite Hopf family of algebras in contrast to the one
containing only finite number of algebras introduced by KLP. Bosonic
representation for A_{\hbar,\eta}(\hat{g}) at level 1 is obtained, and it is
shown that, by repeated application of Drinfeld-like comultiplications, a
realization of A_{\hbar,\eta}(\hat{g}) at any positive integer level can be
obtained. For the special case of g=sl_{r+1}, (r+1)-dimensional evaluation
representation is given. The corresponding intertwining operators are defined
and the intertwining relations are also derived explicitly.


title: Cauchy identities for universal Schubert polynomials

abstract:
  We prove the Cauchy type identities for the universal double Schubert
polynomials, introduced recently by W. Fulton. As a corollary, the
determinantal formulae for some specializations of the universal double
Schubert polynomials corresponding to the Grassmannian permutations are
obtained. We also introduce and study the universal Schur functions and
multiparameter deformation of Schubert polynomials.


title: Highest weight representations of the quantum algebra U_h(gl_\infty)

abstract:
  A class of highest weight irreducible representations of the quantum algebra
U_h(gl_\infty) is constructed. Within each module a basis is introduced and the
transformation relations of the basis under the action of the Chevalley
generators are explicitly written.


title: Introduction to quantum groups

abstract:
  We give an elementary introduction to the theory of algebraic and topological
quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall
the basic facts from Hopf (*-) algebra theory, theory of compact (matrix)
quantum groups and the theory of their actions on compact quantum spaces. We
also provide the most important examples, including the classification of
quantum SL(2)-groups, their real forms and quantum spheres. We also consider
quantum SL_q(N)-groups and quantum Lorentz groups.


title: Jones-Wassermann Subfactors for Disconnected Intervals

abstract:
  We show that the Jones-Wassermann subfactors for disconnected intervals,
which are constructed from the representations of loop groups of type $A$, are
finite-depth subfactors. The index value and the dual principal graphs of these
subfactors are completely determined. The square root of the index value in the
case of two disjoint intervals for vacuum representation is the same as the
Quantum 3-manifold invariant of type $A$ evaluated on $S^1\times S^2$.


title: SL(2,Z)-Invariant Spaces Spanned by Modular Units

abstract:
  Characters of rational vertex operator algebras (RVOAs) arising in
2-dimensional conformal field theories often belong (after suitable
normalization) to the (multiplicative) semigroup E^+ of modular units whose
Fourier expansions are in 1+q Z_{>=0}[[q]], up to a fractional power of q. If
even all characters of a RVOA share this property then we have an example of
what we call modular sets, i.e. finite subsets of E^+ whose elements
(additively) span a vector space which is invariant under the usual action of
SL(2,Z). The classification of modular sets and RVOAs seem to be closely
related. In this article we give an explicit description of the group of
modular units generated by E^+, we prove a certain finiteness result for
modular sets contained in a natural semi-subgroup E_* of E^+, and we discuss
consequences.


title: Elliptic quantum groups and Ruijsenaars models

abstract:
  We construct symmetric and exterior powers of the vector representation of
the elliptic quantum groups $E_{\tau,\eta}(gl_N)$. The corresponding transfer
matrices give rise to various integrable difference equations which could be
solved in principle by the nested Bethe ansatz method. In special cases we
recover the Ruijsenaars systems of commuting difference operators.


title: Non-standard quantum so(3,2) and its contractions

abstract:
  A full (triangular) quantum deformation of so(3,2) is presented by
considering this algebra as the conformal algebra of the 2+1 dimensional
Minkowskian spacetime. Non-relativistic contractions are analysed and used to
obtain quantum Hopf structures for the conformal algebras of the 2+1 Galilean
and Carroll spacetimes. Relations between the latter and the null-plane quantum
Poincar\'e algebra are studied.


title: Quantum groups and representations with highest weight

abstract:
  We consider a special category of Hopf algebras, depending on parameters
$\Sigma$ which possess properties similar to the category of representations of
simple Lie group with highest weight $\lambda$. We connect quantum groups to
minimal objects in this categories---they correspond to irreducible
representations in the category of representations with highest weight
$\lambda$. Moreover, we want to correspond quantum groups only to finite
dimensional irreducible representations. This gives us a condition for
$\lambda$: $\lambda$--- is dominant means the minimal object in the category of
representations with highest weight $\lambda$ is finite dimensional. We put
similar condition for $\Sigma$. We call $\Sigma$ dominant if the minimal object
in corresponding category has polynomial growth. Now we propose to define
quantum groups starting from dominant parameters $\Sigma$.


title:
Generalized Rationality and a "Jacobi Identity" for Intertwining
  Operator Algebras

abstract:
  We prove a generalized rationality property and a new identity that we call
the ``Jacobi identity'' for intertwining operator algebras. Most of the main
properties of genus-zero conformal field theories, including the main
properties of vertex operator algebras, modules, intertwining operators,
Verlinde algebras, and fusing and braiding matrices, are incorporated into this
identity. Together with associativity and commutativity for intertwining
operators proved by the author in \cite{H2} and \cite{H3}, the results of the
present paper solve completely the problem of finding a natural purely
algebraic structure on the direct sum of all inequivalent irreducible modules
for a suitable vertex operator algebra. Two equivalent definitions of
intertwining operator algebra in terms of this Jacobi identity are given.


title:
Lie Bialgebra Structures on Twodimensional Galilei Algebra and their
  Lie-Poisson Counterparts

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


title:
Quantum Hall Effect Wave Functions as Cyclic Representations of
  U_q(sl(2))

abstract:
  Quantum Hall effect wave functions corresponding to the filling factors
1/2p+1, 2/2p+1, ..., 2p/2p+1, 1, are shown to form a basis of irreducible
cyclic representation of the quantum algebra U_q(sl(2)) at q^{2p+1}=1. Thus,
the wave functions \Psi_{P/Q} possessing filling factors P/Q<1 where Q is odd
and P, Q are relatively prime integers are classified in terms of U_q(sl(2)).


title:
Drinfeld-Sokolov reduction for difference operators and deformations of
  W-algebras I. The case of Virasoro algebra

abstract:
  We propose a q-difference version of the Drinfeld-Sokolov reduction scheme,
which gives us q-deformations of the classical W-algebras by reduction from
Poisson-Lie loop groups. We consider in detail the case of SL(2). The
nontrivial consistency conditions fix the choice of the classical r-matrix
defining the Poisson-Lie structure on the loop group LSL(2), and this leads to
a new elliptic classical r-matrix. The reduced Poisson algebra coincides with
the deformation of the classical Virasoro algebra previously defined in
q-alg/9505025. We also consider a discrete analogue of this Poisson algebra. In
the second part (q-alg/9702016) the construction is generalized to the case of
an arbitrary semisimple Lie algebra.


title: Cartan-Weyl Basis for Yangian Double $DY(sl_3)$

abstract:
  We give a new realization of $Y(sl_3)$ via Cartan-Weyl elements. An algebraic
description of Yangian Double $DY(sl_3)$, explicit comultiplication formulas
and universal R-matrix are obtained in these terms.


title: The Reversed q-Exponential Functional Relation

abstract:
  After obtaining some useful identities, we prove an additional functional
relation for $q$ exponentials with reversed order of multiplication, as well as
the well known direct one in a completely rigorous manner.


title: Screenings and a universal Lie-de Rham cocycle

abstract:
  Feigin and Fuchs have given a well-known construction of intertwining
operators between "Fock-type" modules over the Virasoro algebra. The
intertwiners are obtained via contour integration of certain "screening
operators" over top homology classes of a configuration space. The main
observation of the present paper is that the screening operators contain more
information. Specifically, at the chain level, the screening operators provide
a certain canonical cocycle of the Virasoro (resp. affine Kac-Moody) algebra
with coefficients in the de Rham complex of an operator-valued local system on
the configuration space. This way we obtain canonical morphisms from higher
homology groups of the above local systems to appropriate higher Ext-groups
between the Fock space representations. Our construction is motivated by, and
in a special case reduces to the construction of Bowknegt et al, see [BMP1],
[BMP2].


title: Quantum Homogeneous Spaces and Coalgebra Bundles

abstract:
  It is shown that quantum homogeneous spaces of a quantum group H can be
viewed as fibres of quantum fibrations with the total space H that are dual to
coalgebra bundles. As concrete examples of such structures the fibrations with
the quantum 2-sphere and the quantum hyperboloid fibres are considered.


title: Bilinear generating functions for orthogonal polynomials

abstract:
  Using realisations of the positive discrete series representations of the Lie
algebra su(1,1) in terms of Meixner-Pollaczek polynomials, the action of
su(1,1) on Poisson kernels of these polynomials is considered. In the tensor
product of two such representations, two sets of eigenfunctions of a certain
operator can be considered and they are shown to be related through continuous
Hahn polynomials. As a result, a bilinear generating function for continuous
Hahn polynomials is obtained involving the Poisson kernel of Meixner-Pollaczek
polynomials. For the positive discrete series representations of the quantised
universal enveloping algebra Uq(su(1,1)) a similar analysis is performed and
leads to a bilinear generating function for Askey-Wilson polynomials involving
the Poisson kernel of Al-Salam and Chihara polynomials.


title:
On Denominators of the Kontsevich Integral and the Universal
  Perturbative Invariant of 3-Manifolds

abstract:
  The integrality of the Kontsevich integral and perturbative invariants is
discussed. We show that the denominator of the degree $n$ part of the
Kontsevich integral of any knot or link is a divisor of $(2!3!... n!)^4(n+1)!$.
We also show that the denominator of of the degree $n$ part of the universal
perturbative invariant of homology 3-spheres is not divisible by any prime
greater than $2n+1$.


title:
Note on the Algebra of Screening Currents for the Quantum Deformed
  W-Algebra

abstract:
  With slight modifications in the zero modes contributions, the positive and
negative screening currents for the quantum deformed W-algebra W_{q,p}(g) can
be put together to form a single algebra which can be regarded as an elliptic
deformation of the universal enveloping algebra of \hat{g}, where g is any
classical simply-laced Lie algebra.


title:
Configuration Space Integrals and Universal Vassiliev Invariants over
  Closed Surfaces

abstract:
  We show the existence of a universal Vassiliev invariant for links in closed
surface cylinders by explicit construction using configuration space integrals.


title: Semi-Infinite Induction and Wakimoto Modules

abstract:
  The purpose of this paper is to suggest the construction and study properties
of semi-infinite induction, which relates to semi-infinite cohomology the same
way induction relates to homology and coinduction to cohomology. We prove a
version of the Shapiro Lemma, relating the semi-infinite cohomology of a module
with that of the semi-infinitely induced module. A practical outcome of our
construction is a simple construction of Wakimoto modules, highest-weight
modules used in double-sided BGG resolutions of irreducible modules.


title:
Alg\`ebre homologique des N-complexes et homologie de Hochschild aux
  racines de l'unit\'e

abstract:
  We set up a homological algebra for N-complexes, which are graded modules
together with a degree -1 endomorphism d satisfying d^N=0. We define Tor- and
Ext-groups for N-complexes and we compute them in terms of their classical
counterparts (N=2). As an application, we get an alternative definition of the
Hochschild homology of an associative algebra out of an N-complex whose
differential is based on a primitive N-th root of unity.


title:
Some IHX type relations on trivalent graphs and symplectic
  representation theory

abstract:
  We consider several algebras that arise in the study of the mapping class
group (by means of topology and Hodge theory) and describe their
symplectic-invariant parts in terms of algebras on trivalent graphs.


title: On some quadratic algebras

abstract:
  We study some quadratic algebras which are appeared in the low-dimensional
topology and Schubert calculus. We introduce the Jucys-Murphy elements in the
braid algebra and in the pure braid group, as well as the Dunkl elements in the
extended affine braid group. Relationships between the Dunkl elements, Dunkl
operators and Jucys-Murphy elements are described.


title: Finite type invariants III: manifold weight systems

abstract:
  We introduce the notion of weight system for finite type invariants of
integral homology 3-spheres, and we show that invariants of type m are
determined, modulo invariants of type m-1, by their associated weight system.


title: The Noncommutative Inhomogeneous Hopf Algebra

abstract:
  From the bicovariant first order differential calculus on inhomogeneous Hopf
algebra ${\cal B}$ we construct the set of right-invariant Maurer-Cartan
one-forms considered as a right-invariant basis of a bicovariant ${\cal
B}$-bimodule over which we develop the Woronowicz's general theory of
differential calculus on quantum groups. In this formalism, we introduce
suitable functionals on ${\cal B}$ which control the inhomogeneous commutation
rules. In particular we find that the homogeneous part of commutation rules
between the translations and those between the generators of the homogeneous
part of ${\cal B}$ and translations are controled by different R-matrices
satisfying nontrivial characteristic equations.


title: Separation of variables and integral relations for special functions

abstract:
  We show that the method of separation of variables gives a natural
generalisation of integral relations for classical special functions of one
variable. The approach is illustrated by giving a new proof of the
``quadratic'' integral relations for the continuous q-ultraspherical
polynomials. The separating integral operator M expressed in terms of the
Askey-Wilson operator is studied in detail: apart from writing down the
characteristic (``separation'') equations it satisfies, we find its spectrum,
eigenfunctions, inversion, invariants (invariant q-difference operators), and
give its interpretation as a fractional q-integration operator. We also give
expansions of the A1 Macdonald polynomials into the eigenfunctions of the
separating operator M and vice versa.


title: Quantization of Poisson structures on $\R^2$

abstract:
  An 'isomorphism' between the 'moduli space' of star products on $\R^2$ and
the 'moduli space' of all formal Poisson structures on $\R^2$ is established.


title: The space of 3-manifolds and Vassiliev finite-type invariants

abstract:
  In this paper we develop the theory of finite-type invariants for
homologically nontrivial 3-manifolds. We construct an infinite-dimensional
affine space with a hypersurface in it corresponding to manifolds with Morse
singularities. Connected components of the complement of this hypersurface
correspond to homeomorphism type of spin 3-manifolds. This suggests the natural
axiomatics of Vassiliev finite-type invariants for arbitrary closed
3-manifolds. An example of an invariant of order 1 is given.


title: An Introduction to n-Categories

abstract:
  An n-category is some sort of algebraic structure consisting of objects,
morphisms between objects, 2-morphisms between morphisms, and so on up to
n-morphisms, together with various ways of composing them. We survey various
concepts of n-category, with an emphasis on `weak' n-categories, in which all
rules governing the composition of j-morphisms hold only up to equivalence. (An
n-morphism is an equivalence if it is invertible, while a j-morphism for j < n
is an equivalence if it is invertible up to a (j+1)-morphism that is an
equivalence.) We discuss applications of weak n-categories to various subjects
including homotopy theory and topological quantum field theory, and review the
definition of weak n-categories recently proposed by Dolan and the author.


title:
Symmetric functions and the Yangian decomposition of the Fock and Basic
  modules of the affine Lie algebra \hat{sl(N)}

abstract:
  The decompositions of the Fock and Basic modules of the affine Lie algebra
\hat{sl(N)} into irreducible submodules of the Yangian algebra Y(gl(N)) are
constructed. Each of the irreducible submodules admits the unique up to
normalization eigenbasis of the maximal commutative subalgebra of the Yangian.
The elements of this eigenbasis are identified with specializations of
Macdonald symmetric functions where both parameters of these functions approach
an N-th primitive root of unity.


title: On Skein Algebras And Sl_2(C)-Character Varieties

abstract:
  This paper gives insight into intriguing connections between two apparently
unrelated theories: the theory of skein modules of 3-manifolds and the theory
of representations of groups into special linear groups of 2 by 2 matrices.
  Let R be a ring with an invertible element A. For any 3-manifold M one can
assign an R-module called the Kauffman bracket skein module of M. If A^2=1 then
this module has a structure of an R-algebra. We investigate this structure and,
in particular, we prove that if R is the field of complex numbers then this
algebra is isomorphic to the (unreduced) coordinate ring of the SL_2-character
variety of pi_1(M). Using that result we develop a theory of Sl_2-character
varieties by use of topological methods.
  We also assign to any surface a relative Kauffman bracket skein algebra. We
prove several results about this non-commutative algebra.
  Our work should be considered in the context of the book of Brumfiel and
Hilden `SL(2) Representations of Finitely Presented Groups,' Cont. Math 187. In
particular we give a topological interpretation to algebraic objects considered
in that book.


title:
Poisson structures on the center of the elliptic algebra
  A_{p,q}(sl(2)_c)

abstract:
  It is shown that the elliptic algebra A_{p,q}(sl(2)_c) has a non trivial
center at the critical level $c=-2$, generalizing the result of Reshetikhin and
Semenov-Tian-Shansky for trigonometric algebras. A family of Poisson structures
indexed by a non-negative integer $k$ is constructed on this center.


title:
BRST Algebra Quantum Double and Quantization of the Proper Time
  Cotangent Bundle

abstract:
  The quantum double for the quantized BRST superalgebra is studied. The
corresponding R-matrix is explicitly constucted. The Hopf algebras of the
double form an analytical variety with coordinates described by the canonical
deformation parameters. This provides the possibility to construct the
nontrivial quantization of the proper time supergroup cotangent bundle. The
group-like classical limit for this quantization corresponds to the generic
super Lie bialgebra of the double.


title: On the tensor product construction for q-differential algebras

abstract:
  We show that for $q\not=-1$ the q-graded tensor product fails to preserve the
q-differential structure of the product algebra and therefore there is no
natural tensor product construction for q-differential algebras.


title: The Affine q-Schur algebra

abstract:
  We introduce an analogue of the $q$-Schur algebra associated to Coxeter
systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The
first construction realizes the algebra as a certain endomorphism algebra
arising from an affine Hecke algebra of type $\hat A_{r-1}$, where $n \geq r$.
This generalizes the original $q$-Schur algebra as defined by Dipper and James,
and the new algebra contains the ordinary $q$-Schur algebra and the affine
Hecke algebra as subalgebras. Using this approach we can prove a double
centralizer property. The second construction realizes the affine $q$-Schur
algebra as the faithful quotient of the action of a quantum group on the tensor
power of a certain module, analogous to the construction of the ordinary
$q$-Schur algebra as a quotient of $U(\frak g \frak l_n)$.


title:
Geometry of Quantum Homogeneous Vector Bundles and Representation Theory
  of Quantum Groups I

abstract:
  Quantum homogeneous vector bundles are introduced by a direct description of
their sections in the context of Woronowicz type compact quantum groups. The
bundles carry natural topologies inherited from the quantum groups, and their
sections furnish projective modules over algebras of functions on quantum
homogeneous spaces. Further properties of the quantum homogeneous vector
bundles are investigated, and their applications to the representation theory
of quantum groups are explored. In particular, quantum Frobenius reciprocity
and a generalized Borel-Weil theorem are established.


title:
Monodromy of solutions of the elliptic quantum
  Knizhnik-Zamolodchikov-Bernard difference equations

abstract:
  The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference
equations associated to the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is a
system of difference equations with values in a tensor product of
representations of the quantum group and defined in terms of the elliptic
R-matrices associated with pairs of representations of the quantum group. In
this paper we solve the qKZB equations in terms of elliptic hypergeometric
functions and decribe the monodromy properties of solutions. It turns out that
the monodromy transformations of solutions are described in terms of elliptic
R-matrices associated with pairs of representations of the "dual" elliptic
quantum group $E_{p,\eta}(sl_2)$, where $p$ is the step of the difference
equations. Our description of the monodromy is analogous to the Kohno-Drinfeld
description the monodromy group of solutions of the KZ differential equations
associated to a simple Lie algebra in terms of the corresponding quantum group.


title: L operators and Drinfeld's generators

abstract:
  Utilizing the multiplicative formula of universal R matrix, the
correspondence between the L operators and Drinfeld's generators is explicitly
calculated for quantum group U_q(g) with g=A_l^{(1)}, B_l^{(1)}, C_l^{(1)},
D_l^{(1)}.


title: Coloured Hopf algebras

abstract:
  Quite recently, a ``coloured'' extension of the Yang-Baxter equation has
appeared in the literature and various solutions of it have been proposed. In
the present contribution, we introduce a generalization of Hopf algebras, to be
referred to as coloured Hopf algebras, wherein the comultiplication, counit,
and antipode maps are labelled by some colour parameters. The latter may take
values in any finite, countably infinite, or uncountably infinite set. A
straightforward extension of the quasitriangularity property involves a
coloured universal ${\cal R}$-matrix, satisfying the coloured Yang-Baxter
equation. We show how coloured Hopf algebras can be constructed from standard
ones by using an algebra isomorphism group, called colour group. Finally, we
present two examples of coloured quantum universal enveloping algebras.


title: Quantum Group Duality and the Cuntz Algebra

abstract:
  The Cuntz algebra carries in a natural way the structure of a module algebra
over the quantized universal enveloping algebra $U_q(g)$, and the structure of
a co-module algebra over the quantum group $G_q$ associated with $U_q(g)$.
These two algebraic structures are dual to each other via the duality between
$G_q$ and $U_q(g)$.


title: Quantization of Teichm\"uller spaces and the quantum dilogarithm

abstract:
  The Teichm\"uller space of punctured surfaces with the Weil-Petersson
symplectic structure and the action of the mapping class group is realized as
the Hamiltonian reduction of a finite dimensional symplectic space where the
mapping class group acts by symplectic rational transformations. Upon
quantization the corresponding (projective) representation of the mapping class
group is generated by the quantum dilogarithms.


title: Coloured Hopf algebras

abstract:
  Some new algebraic structures related to the coloured Yang-Baxter equation,
and termed coloured Hopf algebras, are reviewed. Coloured quantum universal
enveloping algebras of Lie algebras are defined in this context. An extension
to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is
also presented. The coloured two-parameter quantum universal enveloping algebra
of $gl(1/1)$ is considered as an example.


title:
Universal Enveloping Algebra and Differential Calculi on Orthogonal
  q-groups

abstract:
  We review the construction of the multiparametric quantum group
$ISO_{q,r}(N)$ as a projection from $SO_{q,r}(N+2) $ and show that it is a
bicovariant bimodule over $SO_{q,r}(N)$. The universal enveloping algebra
$U_{q,r}(iso(N))$, characterized as the Hopf algebra of regular functionals on
$ISO_{q,r}(N)$, is found as a Hopf subalgebra of $U_{q,r}(so(N+2))$ and is
shown to be a bicovariant bimodule over $U_{q,r}(so(N))$. An R-matrix
formulation of $U_{q,r}(iso(N))$ is given and we prove the pairing
$U_{q,r}(iso(N))\leftrightarrow ISO_{q,r}(N)$. We analyze the subspaces of
$U_{q,r}(iso(N))$ that define bicovariant differential calculi on
$ISO_{q,r}(N)$.


title: Classification of the quantum deformation of the superalgebra $GL(1|1)$

abstract:
  We present a classification of the possible quantum deformations of the
supergroup $GL(1|1)$ and its Lie superalgebra $gl(1|1)$. In each case, the
(super)commutation relations and the Hopf structures are explicitly computed.
For each $R$ matrix, one finds two inequivalent coproducts whether one chooses
an unbraided or a braided framework while the corresponding structures are
isomorphic as algebras. In the braided case, one recovers the classical algebra
$gl(1|1)$ for suitable limits of the deformation parameters but this is no
longer true in the unbraided case.


title:
Poincar\'{e} series of the Weyl group of elliptic Lie algebras
  $A^{(1,1)}_1$ and $A^{(1,1) \ast}_1$

abstract:
  In this note, we calculate the Poincar\'{e} series of elliptic Weyl groups of
type $A^{(1,1)}_1$ and $A^{(1,1) \ast}_1$ and show that they have a simple
expression.


title: Quasideterminants, I

abstract:
  Our experience shows that dealing with noncommutative objects one should not
imitate the classical commutative mathematics, but follow "the way it is"
starting with basics. In this paper we consider mainly two such problems:
noncommutative Pl\"ucker coordinates (as a background of a noncommutative
geometry) and noncommutative Bezout and Vieta theorems (as a background of
noncommutative algebra). We apply these results to the theory of noncommutative
symmetric functions started by Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon.
We also continue our investigation of noncommutative continued fractions and
almost triangular matrices. It turns out that this problem is related with a
computation of quantum cohomology.


title: Jordanian $U_{h,s}gl(2)$ and its coloured realization

abstract:
  A two-parametric non-standard (Jordanian) deformation of the Lie algebra
$gl(2)$ is constructed, and then, exploited to obtain a new, triangular
R-matrix solution of the coloured Yang-Baxter equation. The corresponding
coloured quantum group is presented explicitly.


title: Duality for the Jordanian Matrix Quantum Group $GL_{g,h}(2)$

abstract:
  We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group
$GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters
and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$
(with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of
$U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra
structure depends on both parameters. We discuss also two one-parameter special
cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and
coincides with the algebra introduced by Ohn as the dual of $SL_h(2)$. The
subalgebra $U'_{-h,h}$ is isomorphic to $U(sl(2))$ as an algebra but has a
nontrivial coalgebra structure and again is not a Hopf subalgebra of
$U_{-h,h}$.


title: A note on the eight tetrahedron equations

abstract:
  In this paper we derive from arguments of string scattering a set of eight
tetrahedron equations, with different index orderings. It is argued that this
system of equations is the proper system that represents integrable structures
in three dimensions generalising the Yang-Baxter equation. Under additional
restrictions this system reduces to the usual tetrahedron equation in the
vertex form. Most known solutions fall under this class, but it is by no means
necessary. Comparison is made with the work on braided monoidal 2-categories
also leading to eight tetrahedron equations.


title: Coloured quantum universal enveloping algebras

abstract:
  We define some new algebraic structures, termed coloured Hopf algebras, by
combining the coalgebra structures and antipodes of a standard Hopf algebra set
$\cal H$, corresponding to some parameter set $\cal Q$, with the
transformations of an algebra isomorphism group $\cal G$, herein called colour
group. Such transformations are labelled by some colour parameters, taking
values in a colour set $\cal C$. We show that various classes of Hopf algebras,
such as almost cocommutative, coboundary, quasitriangular, and triangular ones,
can be extended into corresponding coloured algebraic structures, and that
coloured quasitriangular Hopf algebras, in particular, are characterized by the
existence of a coloured universal $\cal R$-matrix, satisfying the coloured
Yang-Baxter equation. The present definitions extend those previously
introduced by Ohtsuki, which correspond to some substructures in those cases
where the colour group is abelian. We apply the new concepts to construct
coloured quantum universal enveloping algebras of both semisimple and
nonsemisimple Lie algebras, considering several examples with fixed or varying
parameters. As a by-product, some of the matrix representations of coloured
universal $\cal R$-matrices, derived in the present paper, provide new
solutions of the coloured Yang-Baxter equation, which might be of interest in
the context of integrable models.


title: On the affine Temperley-Lieb algebras

abstract:
  We describe the cell structure of the affine Temperley-Lieb algebra with
respect to a monomial basis. We construct a diagram calculus for this algebra.


title:
The Aarhus integral of rational homology 3-spheres I: A highly non
  trivial flat connection on S^3

abstract:
  Path integrals don't really exist, but it is very useful to dream that they
do exist, and figure out the consequences. Apart from describing much of the
physical world as we now know it, these dreams also lead to some highly
non-trivial mathematical theorems and theories. We argue that even though
non-trivial flat connections on S^3 don't really exist, it is beneficial to
dream that one exists (and, in fact, that it comes from the non-existent
Chern-Simons path integral). Dreaming the right way, we are led to a rigorous
construction of a universal finite-type invariant of rational homology spheres.
We show that this invariant is equal to the LMO (Le-Murakami-Ohtsuki) invariant
and that it recovers the Rozansky and Ohtsuki invariants.
  This is part I of a 4-part series, containing the introductions and answers
to some frequently asked questions. Theorems are stated but not proved in this
part, and it can be viewed as a "research announcement". Part II of this series
is titled "Invariance and Universality" (see math/9801049), part III "The
Relation with the Le-Murakami-Ohtsuki Invariant" (see math/9808013), and part
IV "The Relation with the Rozansky and Ohtsuki Invariants".


title: On Vassiliev Invariants not Coming from Semisimple Lie Algebras

abstract:
  We prove a refinement of Vogel's statement that the Vassiliev invariants of
knots coming from semisimple Lie algebras do not generate all Vassiliev
invariants. This refinement takes into account the second grading on Vassiliev
invariants induced by cabling of knots. As an application we get an
amelioration of the actually known lower bounds for the dimensions of the space
of Vassiliev invariants.


title:
Multivariable Al-Salam & Carlitz polynomials associated with the type A
  q-Dunkl kernel

abstract:
  The Al-Salam & Carlitz polynomials are $q$-generalizations of the classical
Hermite polynomials. Multivariable generalizations of these polynomials are
introduced via a generating function involving a multivariable hypergeometric
function which is the $q$-analogue of the type-$A$ Dunkl integral kernel. An
eigenoperator is established for these polynomials and this is used to prove
orthogonality with respect to a certain Jackson integral inner product. This
inner product is normalized by deriving a $q$-analogue of the Mehta integral,
and the corresponding normalization of the multivariable Al-Salam & Carlitz
polynomials is derived from a Pieri-type formula. Various other special
properties of the polynomials are also presented, including their relationship
to the shifted Macdonald polynomials and the big $q$-Jacobi polynomials.


title: Quasitriangular structures on cocommutative Hopf algebras

abstract:
  The article is devoted to the describtion of quasitriangular structures
(universal R-matrices) on cocommutative Hopf algebras. It is known that such
structures are concentrated on finite dimensional Hopf subalgebras. In
particular, quasitriangular structure on group algebra is defined by the pairs
of normal inclusions of an finite abelian group and by invariant
bimultiplicative form on it. The structure is triangular in the case of
coinciding inclusions and skewsymmetric form. The nonstandart
$\lambda$-structure on the representation ring of finite group, corresponding
to the triangular structure on group ring, is described.


title: Vertex algebras

abstract:
  In this paper we try to define the higher dimensional analogues of vertex
algebras. In other words we define algebras which we hope have the same
relation to higher dimensional quantum field theories that vertex algebras have
to one dimensional quantum field theories (or to ``chiral halves'' of two
dimensional quantum field theories). The main idea is to define "vertex
groups". Then classical vertex algebras turn out to be the same as "associative
commutative algebras" over the simplest nontrivial example of a vertex group.
We investigate commutative algebras over higher dimensional vertex groups, some
of which seem to be closely related to (free) quantum field theories.


title: q-Deformation of the Krichever-Novikov Algebra

abstract:
  The recent focus on deformations of algebras called quantum algebras can be
attributed to the fact that they appear to be the basic algebraic structures
underlying an amazingly diverse set of physical situations. To date many
interesting features of these algebras have been found and they are now known
to belong to a class of algebras called Hopf algebras [1]. The remarkable
aspect of these structures is that they can be regarded as deformations of the
usual Lie algebras. Of late, there has been a considerable interest in the
deformation of the Virasoro algebra and the underlying Heisenberg algebra
[2-11]. In this letter we focus our attention on deforming generalizations of
these algebras, namely the Krichever-Novikov (KN) algebra and its associated
Heisenberg algebra.


title: Inverse Harish-Chandra Transform and Difference Operators

abstract:
  We apply a new technique based on double affine Hecke algebras to the
Harish-Chandra theory of spherical zonal functions. The formulas for the
Fourier transforms of the multiplications by the coordinates are obtained as
well as a simple proof of the Harish-Chandra inversion theorem using the Opdam
transform.


title:
The decomposition of level-1 irreducible highest weight modules with
  respect to the level-0 actions of the quantum affine algebra
  $U'_q(\hat{sl}_n)$

abstract:
  We decompose the level-1 irreducible highest weight modules of the quantum
affine algebra $U_q(\hat{sl}_n)$ with respect to the level-0 $U'_q
(\hat{sl}_n)$--action defined in q-alg/9702024. The decomposition is
parameterized by the skew Young diagrams of the border strip type.


title:
An analogue of the Hom functor and a generalized nuclear democracy
  theorem

abstract:
  We give an analogue of the Hom functor and prove a generalized form of the
nuclear democracy theorem of Tsuchiya and Kanie by using a notion of tensor
product for two modules for a vertex operator algebra.


title:
New ${\cal W}_{q,p}(sl(2))$ algebras from the elliptic algebra ${\cal
  A}_{q,p}({\hat sl}(2)_c)$

abstract:
  We construct operators t(z) in the elliptic algebra introduced by Foda et al.
${\cal A}_{q,p}({\hat sl}(2)_c)$. They close an exchange algebra when
p^m=q^{c+2} for m integer. In addition they commute when p=q^{2k} for k integer
non-zero, and they belong to the center of ${\cal A}_{q,p}({\hat sl}(2)_c)$
when k is odd. The Poisson structures obtained for t(z) in these classical
limits are identical to the q-deformed Virasoro Poisson algebra, characterizing
the exchange algebras at generic values of p, q and m as new ${\cal
W}_{q,p}(sl(2))$ algebras.


title: A Jordanian quantum two-photon/Schrodinger algebra

abstract:
  A non-standard quantum deformation of the two-photon algebra $h_6$ is
constructed, and its quantum universal R-matrix is given. Representations of
this new quantum algebra are studied on the Fock space and translated into
Fock-Bargmann realizations that provide a direct formalism for the definition
of deformed states of light. Finally, the isomorphism between $h_6$ and the
(1+1) Schr\"odinger algebra is used to introduce a new (non-standard) Hopf
algebra deformation of this latter symmetry algebra.


title:
Bosonization of quantum affine groups and its application to the higher
  spin Heisenberg model

abstract:
  In this paper, we present a detailed analysis of the diagonalization of the
higher spin Heisenberg model using its quantum affine symmetry
$U_q(\hat{sl(2)})$. In particular, we describe the bosonizations of the latter
algebra, its highest weight representations, vertex operators and screening
operators. Finally, we use this bosonization method to compute the
vacuum-to-vacuum expectation values and the form factors of any local operator.


title: A Poincar\'e-Birkhoff-Witt theorem for generalized color Lie algebras

abstract:
  A proof of Poincar\'e-Birkhoff-Witt theorem is given for a class of
generalized Lie algebras closely related to the Gurevich S-Lie algebras. As
concrete examples, we construct the positive (negative) parts of the quantized
universal enveloping algebras of type A_n and M_{p,q,e}(n,K), which is a
non-standard quantum deformation of GL(n). In particular, we get, for both
algebras, a unified proof of the Poincar\'e-Birkhoff-Witt theorem and we show
that they are genuine universal enveloping algebras of certain generalized Lie
algebras.


title: On classification of dynamical r-matrices

abstract:
  Using recent results of P. Etingof and A. Varchenko on the Classical
Dynamical Yang-Baxter equation, we reduce the classification of dynamical
r-matrices r on a commutative subalgebra l of a Lie algebra g to a purely
algebraic problem under some technical conditions on the symmetric part of r.
Using this, we then classify all non skew-symmetric dynamical r-matrices when g
is a simple Lie algebra and l a commutative subalgebra containing a regular
semisimple element. This partially answers a question in [EV] q-alg/9703040 and
generalizes the Belavin-Drinfeld classification of constant r-matrices. This
classification is similar and in some sense simpler than the Belavin-Drinfeld
classification.


title: A PBW basis for Lusztig's form of untwisted affine quantum groups

abstract:
  Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field
$ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping
algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $
U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley
generators over a suitable subring $ R $ of $ K(q) \, $. We prove a
Poincar\'e-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $,
yielding a basis over $ R $ made of ordered products of $ q $-divided powers of
suitable quantum root vectors.


title:
Intertwining Operators for a Degenerate Double Affine Hecke Algebra and
  Multivariable Orthogonal Polynomials

abstract:
  Operators that intertwine representations of a degenerate version of the
double affine Hecke algebra are introduced. Each of the representations is
related to multi-variable orthogonal polynomials associated with
Calogero-Sutherland type models. As applications, raising operators and shift
operators for such polynomials are constructed.

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


title:
The space of free coherent states is isomorphic to the space of
  distributions on p-adic numbers

abstract:
  Free coherent states for a system with p degrees of freedom are defined. An
isomorphism of the space of free coherent states to the space of distributions
on p-adic disk is constructed.