title:
Self-dual Vertex Operator Superalgebras with Shadows of large minimal
  weight

abstract:
  The shadow V' of a self-dual vertex operator superalgebra V is defined as the
direct sum of the irreducible modules of its even vertex operator subalgebra
$V_{(0)}$ not contained in $V=V_{(0)}+V_{(1)}$. We describe the self-dual
``very nice'' unitary rational vertex operator superalgebras V of rank c whose
shadows have the largest possible minimal weights c/8 or c/8-1. The results are
analogous to and imply the corresponding results for self-dual binary codes and
lattices.


title:
An Alternative to the Chevalley Description of U[sl(n+1)] and
  U_q[sl(n+1)]

abstract:
  An alternative to the Chevalley description of Lie algebra sl(n+1), of its
universal enveloping algebra U[sl(n+1)] and of its q-deformed analogue
U_q[sl(n+1)] in terms of generators, called creation and annihilation
generators (CAGs), and relations is given. It is indicated (without further
discussions) that the representations of the CAGs describe new quantum
statistics, which is a particular case of the Haldane exclusion statistics.


title: Flexible sheaves

abstract:
  We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt,
Mather, Cordier) over a Grothendieck site; we call these ``flexible
presheaves''. After some preliminary materiel, we define the ``flexible sheaf''
condition. This descent condition (known to Thomason) is the same as what
Jardine called being ``flasque'' with respect to the presheaves representable
by objects in the site; and it is more recently known as the condition of being
an $n$-stack. We construct the flexible sheaf associated to a flexible presheaf
in the $n$-truncated case, as an application of a certain natural operation
$n+2$ times. We prove an analogue of Vogt's theorem for the case where the
Grothendieck topology is nontrivial, identifying the set of morphisms in
Illusie's derived category as the set of homotopy classes of homotopy-coherent
morphisms between flexible sheaves. The homotopy-coherent point of view allows
one easily to define the flexible mapping sheaf $Hom (R,T)$ between two
flexible sheaves.
  This revision fills major gaps in the bibliography. References to the
additional items are inserted in the text. A new introduction and abstract are
added (the old ones are retained as comments in the source file). A few other
minor changes in the exposition include arrangement of internal references.


title:
Universal $q$-differential calculus and $q$-analog of homological
  algebra

abstract:
  We recall the definition of $q$-differential algebras and discuss some
representative examples. In particular we construct the $q$-analog of the
Hochschild coboundary. We then construct the universal $q$-differential
envelope of a unital associative algebra and study its properties. The paper
also contains general results on $d^N=0$.


title: The vector k-constrained KP hierarchy and Sato's Grassmannian

abstract:
  We use the representation theory of the infinite matrix group to show that
(in the polynomial case) the $n$--vector $k$--constrained KP hierarchy has a
natural geometrical interpretation on Sato's infinite Grassmannian. This
description generalizes the the $k$--reduced KP or Gelfand--Dickey hierarchies.


title: The $q$-calculus for generic $q$ and $q$ a root of unity

abstract:
  The $q$-calculus for generic $q$ is developed and related to the deformed
oscillator of parameter $q^{1/2}$. By passing with care to the limit in which
$q$ is a root of unity, one uncovers the full algebraic structure of ${{\cal
Z}}_n$-graded fractional supersymmetry and its natural representation.


title: The Many Faces of the Chiral Potts Model

abstract:
  In this talk, we give a brief overview of several aspects of the theory of
the chiral Potts model, including higher-genus solutions of the star-triangle
and tetrahedron equations, cyclic representations of affine quantum groups,
basic hypergeometric functions at root of unity, and possible applications.


title: The topological realization of a simplicial presheaf

abstract:
  The purpose of this article is to define the topological realization of a
simplicial presheaf and to prove (under appropriate conditions) that it is
homotopy-invariant under Illusie weak equivalence. In particular this applies
to the site of schemes over $Spec (\cc)$ with the etale or Zariski topologies.
As an application we show how to calculate the topological realization of a
Deligne-Mumford stack. At the end we speculate on how to extend this to the
case of $n$-topoi.


title:
Implications of the Hopf algebra properties of noncommutative
  differential calculi

abstract:
  We define a noncommutative algebra of four basic objects within a
differential calculus on quantum groups: functions, 1-forms, Lie derivatives
and inner derivations, as the cross-product algebra associated with
Woronowicz's (differential) algebra of functions and forms. This definition
properly takes into account the Hopf algebra structure of the Woronowicz
calculus. It also provides a direct proof of the Cartan identity.


title: Invariance quantum groups of the deformed oscillator algebra

abstract:
  A differential calculus is set up on a deformation of the oscillator algebra.
It is uniquely determined by the requirement of invariance under a
seven-dimensional quantum group. The quantum space and its associated
differential calculus are also shown to be invariant under a nine generator
quantum group containing the previous one.


title: Fusion of q-tensor operators: quasi-Hopf-algebraic point of view

abstract:
  Tensor operators associated with a given quantum Lie algebra admit a natural
description in the R-matrix language. Here we employ the R-matrix approach to
discuss the problem of fusion of tensor operators. The most interesting case is
provided by the quantum WZNW model, where, by construction, we deal with sets
of linearly independent tensor operators. In this case the fusion problem is
equivalent to construction of an analogue F(p) of the twisting element F which
is employed in Drinfeld's description of quasi-Hopf algebras. We discuss the
construction of the twisting element F(p) in a general situation and give
illustrating calculations for the case of the fundamental representation of
U_q(sl(2)).


title:
A three-parameter deformation of the Weyl-Heisenberg algebra:
  differential calculus and invariance

abstract:
  We define a three-parameter deformation of the Weyl-Heisenberg algebra that
generalizes the q-oscillator algebra. By a purely algebraical procedure, we set
up on this quantum space two differential calculi that are shown to be
invariant on the same quantum group, extended to a ten-generator
Hopf-star-algebra. We prove that, when the values of the parameters are
related, the two differential calculi reduce to one that is invariant under two
quantum groups.


title: On the $q$-deformed oscillator algebras:$su_q(1,1)$ and $su_q(2)$

abstract:
  We study the relations between $q$-deformations and $q$-coherent states of
the single oscillator representations for $su_q(1,1)$ and $su_q(2)$ algebras;
Dyson and Holstein-Primakoff type in terms of Biedenharn, Macfarlane and
anyonic oscilators. We also discuss the related Fock-Bargmann $q$-derivative
and integration.


title: First Order Calculi with Values in Right--Universal Bimodules

abstract:
  The purpose of this note is to show how calculi on unital associative algebra
with universal right bimodule generalize previously studied constructions by
Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this
language results are in a natural context, are easier to describe and handle.
As a by--product we obtained intrinsic, coordinate--free and basis--independent
generalization of the first order noncommutative differential calculi with
partial derivatives.


title: Cartan Pairs

abstract:
  A new notion of Cartan pairs as a substitute of notion of vector fields in
noncommutative geometry is proposed. The correspondence between Cartan pairs
and differential calculi is established.


title:
Generalized differential spaces with $d^N=0$ and the $q$-differential
  calculus

abstract:
  We present some results concerning the generalized homologies associated with
nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We
then introduce the notion of graded $q$-differential algebra and describe some
examples. In particular we construct the $q$-analog of the simplicial
differential on forms, the $q$-analog of the Hochschild differential and the
$q$-analog of the universal differential envelope of an associative unital
algebra.


title: Pairing and Quantum Double of Multiplier Hopf Algebras

abstract:
  We define and investigate pairings of multiplier Hopf algebras. It is shown
that two dually paired regular multiplier Hopf ($*$-)algebras $A$ and $B$ yield
a quantum double multiplier Hopf ($*$-)algebra which is again regular.
Integrals on $A$ and $B$ induce an integral on the quantum double. The results
generalize pairing and quantum double construction from ordinary Hopf algebras
to multiplier Hopf algebras.


title: Boundary Solutions of the Classical Yang-Baxter Equation

abstract:
  We define a new class of unitary solutions to the classical Yang-Baxter
equation (CYBE). These ``boundary solutions'' are those which lie in the
closure of the space of unitary solutions to the modified classical Yang-Baxter
equation (MCYBE). Using the Belavin-Drinfel'd classification of the solutions
to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In
particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly
construct for all n > 2 a boundary solution based on the maximal parabolic
subalgebra of sl(n) obtained by deleting the first negative root. We give some
evidence for a generalization of this result pertaining to other maximal
parabolic subalgebras whose omitted root is relatively prime to $n$. We also
give examples of non-boundary solutions for the classical simple Lie algebras.


title: On Witten-type deformation of osp(1/2) Algebra

abstract:
  In this paper Witten type deformation of osp(1/2) algbera is introduced and
its realization and matrix representation are obtained. The matrix
representation is shown to be possible only when the dimension is odd.


title: Two parameter Deformed Multimode Oscillators and q-Symmetric States

abstract:
  Two types of the coherent states for two parameter deformed multimode
oscillator system are investigated. Moreover, two parameter deformed $gl(n)$
algebra and deformed symmetric states are constructed.


title:
$gl_q(n)$-covariant Oscillators and q-Deformed Quantum Mechanics in n
  Dimensions

abstract:
  In this paper the coherent state for $gl_q(n)$-covariant oscillators is
constructed and is shown to satisfy the completeness relation. And the
q-analogue of quantum mechanics in n dimensions is obtained by using $gl_q(n)$
oscillators.


title: Higher-Dimensional Algebra II: 2-Hilbert Spaces

abstract:
  A 2-Hilbert space is a category with structures and properties analogous to
those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an
abelian category enriched over Hilb with a *-structure, conjugate-linear on the
hom-sets, satisfying <fg,h> = <g,f*h> = <f,hg*>. We also define monoidal,
braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we
call 2-H*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we
describe the relation between these and tangles in 2, 3, and 4 dimensions,
respectively. We prove a generalized Doplicher-Roberts theorem stating that
every symmetric 2-H*-algebra is equivalent to the category Rep(G) of continuous
unitary finite-dimensional representations of some compact supergroupoid G. The
equivalence is given by a categorified version of the Gelfand transform; we
also construct a categorified version of the Fourier transform when G is a
compact abelian group. Finally, we characterize Rep(G) by its universal
properties when G is a compact classical group. For example, Rep(U(n)) is the
free connected symmetric 2-H*-algebra on one even object of dimension n.


title: $gl_q(n)$-Covariant Multimode Oscillators and q-Symmetric States

abstract:
  In this paper the coherent states and q-symmetric states for
$gl_q(n)$-covariant multimode oscillator system are investigated.


title: On the Structure of Monodromy Algebras and Drinfeld Doubles

abstract:
  We give a review and some new relations on the structure of the monodromy
algebra (also called loop algebra) associated with a quasitriangular Hopf
algebra H. It is shown that as an algebra it coincides with the so-called
braided group constructed by S. Majid on the dual of H. Gauge transformations
act on monodromy algebras via the coadjoint action. Applying a result of Majid,
the resulting crossed product is isomorphic to the Drinfeld double D(H). Hence,
under the so-called factorizability condition given by N. Reshetikhin and M.
Semenov-Tian- Shansky, both algebras are isomorphic to the algbraic tensor
product H\otimes H. It is indicated that in this way the results of Alekseev et
al. on lattice current algebras are consistent with the theory of more general
Hopf spin chains given by K. Szlach\'anyi and the author. In the Appendix the
multi-loop algebras L_m of Alekseev and Schomerus [AS] are identified with
braided tensor products of monodromy algebras in the sense of Majid, which
leads to an explanation of the ``bosonization formula'' of [AS] representing
L_m as H\otimes\dots\otimes H.


title: Algebraic structure of multi-parameter quantum groups

abstract:
  Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups
U_q(g) and C_q[G] are considered where G is a semi-simple connected complex
algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G]
is calculated, generalizing a result of Joseph for the standard quantum groups.
This classification is compared with the classification of symplectic leaves
for the associated Poisson structure on G.


title:
Noninvertibility and ``Semi-'' Analogs of (Super) Manifolds, Fiber
  Bundles and Homotopies

abstract:
  Supersymmetry contains initially noninvertible objects, but it is common to
deal with the invertible ones only, factorizing former in some extent. We
propose to reconsider this ansatz and try to redefine such fundamental notions
as supermanifolds, fiber bundles and homotopies using some weakening
invertibility conditions. The prefix semi- reflects the fact that the
underlying morphisms form corresponding semigroups consisting of a known group
part and a new ideal noninvertible part. We found that the absence of
invertibility gives us the generalization of the cocycle conditions for
transition functions of supermanifolds and fiber bundles in a natural way,
which can lead to construction of noninvertible analogs of \v{C}ech cocycles.
We define semi-homotopies, which can be noninvertible and describe mappings
into the semi-supermanifolds introduced.

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


title: q-deformed Star Products and Moyal Brackets

abstract:
  The standard and anti-standard ordered operators acting on two-dimensional
q-deformed phase space are shown to satisfy algebras which can be called
W_\infty. q-star products and q-Moyal brackets corresponding to these algebras
are constructed. Some applications like defining q-classical mechanics and
q-path integrals are discussed.


title: Skein Quantization and Lattice Gauge Field Theory

abstract:
  This is a survey article describing the various ways in which the Kauffman
bracket skein module is a quantization of surface group characters. These
include a purely heuristic sense of deformation of a presentation, a Poisson
quantization, and a lattice gauge field theory quantization.


title: Coherent States of $gl_q(2)$-covariant Oscillators

abstract:
  In this paper two types of coherent states of $gl_q(2)$-covariant oscillators
are investigated.


title: On Clifford representation of Hopf algebras and Fierz identities

abstract:
  We present a short review of the action and coaction of Hopf algebras on
Clifford algebras as an introduction to physically meaningful examples. Some
q-deformed Clifford algebras are studied from this context and conclusions are
derived.


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

abstract:
  We apply the theory of finite-type invariants of homology 3-spheres to
investigate the structure of the Torelli group. We construct natural cocycles
in the Torelli group and show that the lower central series quotients of the
Torelli group map onto a vector space of trivalent graphs. We also have
analogous results for two other natural subgroups of the mapping class group.


title: A q-oscillator Green Function

abstract:
  By using the generating function formula for the product of two q-Hermite
polynomials q-deformation of the Feynman Green function for the harmonic
oscillator is obtained.


title: Nonstandard quantum groups associated to Belavin-Drinfeld triples

abstract:
  A construction is given of a family of non-standard quantizations of the
algebra of functions on a connected complex semi-simple algebraic group. For
each ``disjoint'' triple in the sense of Belavin and Drinfeld, a 2-cocycle is
constructed on certain multi-parameter quantum groups. The new non-standard
quantum groups are the Hopf algebras obtained by twisting the known quantum
groups by these 2-cocycles. In particular, the Cremmer-Gervais quantization of
$SL(3)$ can be constructed in this way.


title: Paragrassmann Integral, Discrete Systems and Quantum Groups

abstract:
  Some aspects of differential and integral calculi on generalized grassmann
(paragrassmann) algebras are considered. The integration over paragrassmann
variables is applied to evaluate the partition function for the $Z_{p+1}$ Potts
model on a chain. Finite dimensional paragrassmann representations for
$GL_{q}(2)$ are constructed.


title: Quantum Affine Algebras at Roots of Unity

abstract:
  We study the restricted form of the qaunatized enveloping algebra of an
untwisted affine Lie algebra and prove a triangular decomposition for it. In
proving the decomposition we prove several new identities in the quantized
algebra, one of these show a connection between the quantized algebra and Young
diagrams. These identities are all invisible in the non-quantum case of the
problem which was considered by Garland in 1978. We then study the
finite-dimensional irreducible representations and prove a factorization
theorem for such representations.


title:
Confluent hypergeometric orthogonal polynomials related to the rational
  quantum Calogero system with harmonic confinement

abstract:
  Two families (type $A$ and type $B$) of confluent hypergeometric polynomials
in several variables are studied. We describe the orthogonality properties,
differential equations, and Pieri type recurrence formulas for these families.
In the one-variable case, the polynomials in question reduce to the Hermite
polynomials (type $A$) and the Laguerre polynomials (type $B$), respectively.
The multivariable confluent hypergeometric families considered here may be used
to diagonalize the rational quantum Calogero models with harmonic confinement
(for the classical root systems) and are closely connected to the (symmetric)
generalized spherical harmonics investigated by Dunkl.


title:
Anyonic Realization of the Quantum Affine Lie Superalgebra
  U_q(A(M,N)^{(1)})

abstract:
  We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1)
in terms of anyons defined on a one or two-dimensional lattice, the deformation
parameter q being related to the statistical parameter $\nu$ of the anyons by q
= exp(i\pi\nu). The construction uses anyons contructed from usual fermionic
oscillators and deformed bosonic oscillators. As a byproduct, realization
deformed in any sector of the quantum superalgebras U_q(A(M,N)) is obtained.


title:
Some Realization of $gl_q(n)$-covariant Oscillator Algebra and
  $gl_q(n)$-covariant q-Virasoro Algebra with $q$ a root of unity

abstract:
  In this paper some realization of $gl_q(n)$-covariant oscillators is obtained
when $q$ is a root of unity. And the $gl_q(n)$-covariant q-Virasoro algebra is
presented by using the $gl_q(n)$-covariant oscillators.


title:
Irreducible representations of deformed oscillator algebra and q-special
  functions

abstract:
  Different generators of a deformed oscillator algebra give rise to
one-parameter families of $q$-exponential functions and $q$-Hermite polynomials
related by generating functions. Connections of the Stieltjes and Hamburger
classical moment problems with the corresponding resolution of unity for the
$q$-coherent states and with 'coordinate' operators - Jacobi matrices, are also
pointed out.


title: Advances in Quantum and Braided Geometry

abstract:
  We demonstrate our recent general results on the Casimir construction and
moduli space of all bicovariant calculi by means of some detailed examples,
including finite-difference and 2-jet cacluli on $\R^n$ and full details of the
Casimir construction of the 4D calculus of $SU_q(2)$. We likewise demonstrate
our previous general constructions with T. Brzezinski of quantum group gauge
theory with examples of such nonuniversal differential calculi on spacetime. We
outline a notion of quantum homotopy of a quantum space. We indicate a possible
application to classical integrable systems.


title: New Quantum Groups by Double-Bosonisation

abstract:
  We obtain new family of quasitriangular Hopf algebras $C^{0|n}_q\lcross
\widetilde{U_q(su_n)}\rcross C^{0|n}_q$ via the author's recent
double-bosonisation construction for new quantum groups. They are versions of
$U_q(su_{n+1})$ with a fermionic rather than bosonic quantum plane of roots
adjoined to $U_q(su_n)$. We give the $n=2$ case in detail. We also consider the
anyonic-double of an anyonic ($Z/nZ$-graded) braided group and the
double-bosonisation of the free braided group in $n$ variables.


title:
Deforming Maps for Lie Group Covariant Creation and Annihilation
  Operators

abstract:
  Any deformation of a Weyl or Clifford algebra A can be realized through a
`deforming map', i.e. a formal change of generators in A. This is true in
particular if A is covariant under a Lie algebra g and its deformation is
induced by some triangular deformation $U_h g$ of the Hopf algebra $Ug$. We
propose a systematic method to construct all the corresponding deforming maps,
together with the corresponding realizations of the action of $U_h g$. The
method is then generalized and explicitly applied to the case that $U_h g$ is
the quantum group $U_h sl(2)$. A preliminary study of the status of deforming
maps at the representation level shows in particular that `deformed' Fock
representations induced by a compact $U_h g$ can be interpreted as standard
`undeformed' Fock representations describing particles with ordinary Bose or
Fermi statistics.


title: Induced representations of the one dimensional quantum Galilei group

abstract:
  We study the representations of the quantum Galilei group by a suitable
generalization of the Kirillov method on spaces of non commutative functions.
On these spaces we determine a quasi-invariant measure with respect to the
action of the quantum group by which we discuss unitary and irreducible
representations. The latter are equivalent to representations on \ell^2, i.e.
on the space of square summable functions on a one dimensional lattice.


title: Quantum Heisenberg--Weyl Algebras

abstract:
  All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$
are classified and explicitly quantized. The complete list of quantum
Heisenberg--Weyl algebras so obtained includes new multiparameter deformations,
most of them being of the non-coboundary type.


title: The Polar Decomposition of q-Deformed Boson Algebra

abstract:
  In this paper we used the finite Fourier transformation to obtain the polar
decomposition of the q-deformed boson algebra with $q$ a root of unity.


title: Quantum Dynamical R-matrices and Quantum Frobenius Group

abstract:
  We propose an algebraic scheme for quantizing the rational
Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special
parameterization of the cotangent bundle over GL(N,C). In new variables the
standard symplectic structure is described by a classical (Frobenius) r-matrix
and by a new dynamical $\bar{r}$-matrix. Quantizing both of them we find the
quantum L-operator algebra and construct its particular representation
corresponding to the rational Ruijsenaars-Schneider system. Using the dual
parameterization of the cotangent bundle we also derive the algebra for the
L-operator of the trigonometric Calogero-Moser system.


title: The Quantum Symplectic Cayley-Klein Groups

abstract:
  The contraction method applied to the construction of the nonsemisimple
quantum symplectic Cayley-Klein groups $ Fun(Sp_q(n;j)) $. This groups has been
realised as Hopf algebra of the noncommutative functions over the algebra with
nilpotent generators. The dual quantum algebras $ sp_q(n;j) $ are constructed.


title: Quantum Orthogonal Cayley-Klein Groups in Cartesian Basis

abstract:
  The similarity transformations of quantum orthogonal groups are developed and
FRT theory is reformulated to the Cartesian basis. The quantum orthogonal
Cayley-Klein groups are introduced as the algebra functions over an associative
algebra with the nilpotent generators. The quantum orthogonal Cayley-Klein
algebras are obtained as the dual objects to the corresponding quantum groups.


title: On the Drinfeld Twist for Quantum sl(2)

abstract:
  An isomorphism, up to a twist, between the quasitriangular quantum enveloping
algebra U_h(sl(2)) and the (classical) U(sl(2))[[h]]is discussed. The universal
twisting element $\cal F$ is given up to the second order in the deformation
parameter $h$.


title: Coset construction for winding subalgebras and applications

abstract:
  In this paper we review the coset construction for winding subalgebras of
affine Lie algebras. We classify all cosets of central charge $\hat c<1$ and
calculate their branching rules. The corresponding character identities give
certain `doubling formulae' for the affine characters. We discuss some
applications of our construction, in particular we find a simple proof of a
crucial identity needed for the computation of the level-2 root multiplicities
of the hyperbolic Kac-Moody algebra $E_{10}$.


title: A reproducing kernel for nonsymmetric Macdonald polynomials

abstract:
  We present a new formula of Cauchy type for the nonsymmetric Macdonald
polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the
explicit formula for a reproducing kernel on the polynomial ring of n
variables.


title: Turaev-Viro Modules of Satellite knots

abstract:
  Given an oriented knot K in S^3 and a TQFT, Turaev and Viro defined modules
somewhat analogous to the Alexander module. We work with the (V_p,Z_p) theories
of Blanchet, Habegger, Masbaum and Vogel {BHMV} for p \ge 3, and consider the
associated modules. In {G}, we defined modules which also depend on the extra
data of a color c which is assigned to a meridian of the knot in the
construction of the module. These modules can be used to calculate the quantum
invariants of cyclic branched covers of knots and have other uses. Suppose now
that S is a satellite knot with companion C, and pattern P. We give formulas
for the Turaev-Viro modules for S in terms of the Turaev-Viro modules of C and
similar data coming from the pattern P. We compute these invariants explicitly
in several examples.


title: A recursion and a combinatorial formula for Jack polynomials

abstract:
  Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials
using Cherednik operators. In this paper, we derive a simple recursion formula
for these polynomials and formulas relating the symmetric Jack polynomials with
the non-symmetric ones. These formulas are then implemented by a closed
expression of symmetric and non-symmetric Jack polynomials in terms of certain
tableaux. The main application is a proof of a conjecture of Macdonald stating
certain integrality and positivity properties of Jack polynomials.


title: Difference equations and symmetric polynomials defined by their zeros

abstract:
  In this paper, we introduce a new family of symmetric polynomials which
depends on a parameter r. They are defined by specifying certain of their
zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in
terms of Capelli identities.
  First, we give explicit formulas in some special cases. Then we show that the
polynomials can also be defined in terms of difference equations. As a
corollary we obtain that their top homogeneous part is a Jack polynomial. This
is used to give a new proof of the Pieri formula for Jack polynomials.


title:
A new derivation of the inner product formula for the Macdonald
  symmetric polynomials

abstract:
  We give a short proof of the inner product conjecture for the symmetric
Macdonald polynomials of type $A_{n-1}$. As a special case, the corresponding
constant term conjecture is also proved.


title:
The infinitesimal form of induced representation of the kappa-Poincare
  group

abstract:
  The infinitesimal form of the induced representation of the kappa-Poincare
group is constructed. The infinitesimal action of the kappa-Poincare group on
the kappa-Minkowski space is described. The actions of these two infinitesimal
forms on the solution of Klein-Gordon equation are compared.


title: Semiinfinite cohomology of quantum groups II

abstract:
  It is known that the semi-infinite cohomology spaces of the infinitely
twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in
an integrable simple module over the affine Lie algebra have a base enumerated
by elements of the corresponding affine Weyl group graded by the semiinfinite
length function. Let $U$ be the affine quantum group corresponding to $g$. It
is possible to define a subalgebra in $U$ being the quantum analogue of the
universal enveloping algebra of the infinitely twisted nilpotent subalgebra in
$g$. In this paper we prove that for general values of the parameter $v$ the
semiinfinite cohomology of this associative algebra with coefficients in an
integrable simple module over $U$ coincides with the one of the corresponding
Lie subalgebra in $g$ with coefficients in the corresponding $g$-module.


title: The Twisted Heisenberg Algebra ${\cal U}_{h,w}({\cal H}(4))$

abstract:
  A two parametric deformation of the enveloping Heisenberg algebra ${\cal
H}(4)$ which appear as a combination of the standard and a nonstandard
quantization given by Ballesteros and Herranz is defined and proved to be
Ribbon Hopf algebra. The universal ${\cal R}$-matrix and its associated quantum
group are constructed. New solution of Braid group are obtained. The
contribution of these parameters in invariants of links and WZW model are
analyzed. General results for twisted Ribbon Hopf algebra are derived.


title:
Quantum double Schubert polynomials, quantum Schubert polynomials and
  Vafa-Intriligator formula

abstract:
  We study the algebraic aspects of equivariant quantum cohomology algebra of
the flag manifold. We introduce and study the quantum double Schubert
polynomials, which are the Lascoux-Schutzenberger type representatives of the
equivariant quantum cohomology classes. Our approach is based on the quantum
Cauchy identity. We define also quantum Schubert polynomials as the
Gram-Schmidt orthogonalization of some set of monomials with respect to the
scalar product, defined by the Grothendieck residue. Using quantum Cauchy
identity, we prove that quantum Schubert polynomials are the specialization of
quantum double Schubert polynomials with second set of variables equals to
zero, and as a corollary obtain a simple formula for the quantum Schubert
polynomials. We also prove the higher genus analog of Vafa-Intriligator's
formula for the flag manifolds and study the quantum residues generating
function. We introduce the extended Ehresman-Bruhat order on the symmetric
group and formulate the equivariant quantum Pieri rule.


title:
Difference equations of quantum current operators and quantum
  parafermion construction

abstract:
  For the current realization of the affine quantum groups, a simple
comultiplication for the quantum current operators was given by Drinfeld. With
this comultiplication, we prove that, for the integrable modules of $U_q(\hat
{\frak sl}(2))$ of level $k+1$, $x^\pm(z)x^\pm(zq^{\pm 2}) \cdot\cdot\cdot
x^\pm(zq^{\pm 2k})$ are vertex operators satisfying certain q-difference
equations, and we derive the quantum parafermions of $U_q(\hat {\frak sl}(2))$.


title: Algebraic integrability of the two-body Ruijsenaars operator

abstract:
  We study the algebra of difference operators that commute with the two-body
Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator,
for generic values of the deformation parameter. The algebra is commutative. It
is the algebra of polynomial functions on an affine hyperelliptic curve
$Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue
problem.


title:
Non integrable representations of the restricted quantum analogue of
  sl(3) at roots of 1

abstract:
  The structure of irreducible representations of (restricted) U_q(sl(3)) at
roots of unity is understood within the Gelfand--Zetlin basis. The latter needs
a weakened definition for non integrable representations, where the quadratic
Casimir operator of the quantum subalgebra U_q(sl(2)) of U_q(sl(3)) is not
completely diagonalized. This is necessary in order to take in account the
indecomposable U_q(sl(2))-modules that appear. The set of redefined (mixed)
states has a teepee shape inside the pyramid made with the whole
representation.


title: Topics in hidden symmetries. II

abstract:
  This short paper being devoted to some aspects of the inverse problem of the
representation theory briefly treats the interrelations between the author's
approach to the setting free of hidden symmetries and the researches of
D.P.Zhelobenko in the generalized Mickelsson algebras and related topics.


title: Representations of $U_h(su(N))$ derived from quantum flag manifolds

abstract:
  A relationship between quantum flag and Grassmann manifolds is revealed. This
enables a formal diagonalization of quantum positive matrices. The requirement
that this diagonalization defines a homomorphism leads to a left \Uh -- module
structure on the algebra generated by quantum antiholomorphic coordinate
functions living on the flag manifold. The module is defined by prescribing the
action on the unit and then extending it to all polynomials using a quantum
version of Leibniz rule. Leibniz rule is shown to be induced by the dressing
transformation. For discrete values of parameters occuring in the
diagonalization one can extract finite-dimensional irreducible representations
of \Uh as cyclic submodules.


title: Invariants of fibred knots from moduli

abstract:
  An invariant $\mu_{\alpha}(K)$ of fibred knots $K$ in a homology sphere is
defined for each $\alpha \in {\bold S}{\bold U}_n$ as follows. Since the knot
is fibred, the knot complement is described by an element of the mapping class
group, which induces an action on the variety of ${\bold S}{\bold U}_n$
representations of the surface group. Restricting attention to those
representations with holonomy along the longitude conjugate to $a \in {\bold
S}{\bold U}_n,$ one can define $\mu_{\alpha}(K)$ to be the Lefschetz number of
this action. The dependence of $\mu_\alpha(K)$ on $\alpha$ is studied and
formulas relating $\mu_{\alpha}(K)$ to $\mu_{\beta}(K)$ are derived for
$\alpha,\beta \in
  ${\bold S}{\bold U}_n$.$


title:
Integrality of the Averaged Jones Polynomial of Algebraically Split
  Links

abstract:
  X.-S. Lin and Z. Wang recently made a conjecture concerning the integrality
of the Taylor coefficients of the averaged Jones polynomial of algebraically
split links. This question is related to a conjectural integrality result for
the Ohtsuki invariants, which are rational topological invariants of homology
3-spheres derived from the quantum invariants of Witten and Reshetikhin-Turaev.
  This paper presents counterexamples to the conjecture of Lin and Wang and
gives a proof of a corrected statement of the conjecture. The statement is
divided into two parts. The first part applies to geometrically split links and
gives an integrality result which is stronger than conjectured, while the
second is valid for any algebraically split link and gives a somewhat weaker
statement. Both propositions are seen to be sharp by the examples given.


title: Quantization of Lie bialgebras, III

abstract:
  In this paper we construct explicitly the quantization of Lie bialgebras of
$\g$-valued functions on a punctured rational or elliptic curve, where $\g$ is
a finite dimensional simple Lie algebra. by reducing the problem of
quantization of the algebra of $\g$-valued functions on a curve with many
punctures to the case of one puncture.


title: On ${\cal U}_h(sl(2))$, ${\cal U}_h(e(3))$ and their Representations

abstract:
  By solving a set of recursion relations for the matrix elements of the ${\cal
U}_h(sl(2))$ generators, the finite dimensional highest weight representations
of the algebra were obtained as factor representations. Taking a nonlinear
combination of the generators of the two copies of the ${\cal U}_h(sl(2))$
algebra, we obtained ${\cal U}_h(so(4))$ algebra. The latter, on contraction,
yields ${\cal U}_h(e(3))$ algebra. A nonlinear map of ${\cal U}_h(e(3))$
algebra on its classical analogue $e(3)$ was obtained. The inverse mapping was
found to be singular. It signifies a physically interesting situation, where in
the momentum basis, a restricted domain of the eigenvalues of the classical
operators is mapped on the whole real domain of the eigenvalues of the deformed
operators.


title: Capelli identities for Lie superalgebras

abstract:
  We apply the technique of affine Hecke algebras to the invariant theory of
the "queer" Lie superalgebra $q_N$. We give explicit formulas for the elements
of a distinguished basis in the centre of $U(q_N)$, determined by "vanishing"
conditions. We evaluate the corresponding differential operators on
$C^{N|N}\otimes C^{M|M}$. In particular, we obtain an analogue for $q_N$ of the
classical Capelli identity.


title: Iwahori-Hecke algebras of type A at roots of unity

abstract:
  In this paper, we explore the use of path idempotents for the Hecke algebra
of type $A$ at roots of unity. For $q$ a primitive $\ell$-th root of unity we
obain a non-unital imbedding of (a quotient of) the group algebra of $S_m$ into
(a quotient of) the Hecke algebra $H_n(q)$ for certain $m$ and $n$. From this
we recover certain instances of irreducibility criteria of Dipper, James, and
Mathas, and we derive estimates on the decomposition numbers for the Hecke
algebra at roots of unity. The bounds are easily computed, provide a good
geometric picture of the pairs of diagrams $\lambda$, $\mu$ for which the
decomposition number $d_{\lambda, \mu}$ is non-zero, and also appers to be a
useful adjunct to the exact computation of the decomposition numbers.


title: Quantum Grothendieck polynomials

abstract:
  We study the algebraic aspects of (small) quantum equivariant $K$-theory of
flag manifold. Lascoux-Sch\"utzenberger's type formula for quantum double and
quantum double dual Grothendieck polynomials and the quantum Cauchy identity
for quantum Grothendieck polynomials are obtained.


title: On Drinfeld realization of quantum affine algebras

abstract:
  We provide a direct proof of the Drinfeld realization for the quantum affine
algebras.


title: Singular R-matrices and Drinfeld's comultiplication

abstract:
  We compute the R-matrix which intertwines two dimensional evaluation
representations with Drinfeld comultiplication for U_q(\widehat{sl}_2). This
R-matrix contains terms proportional to the delta-function. We construct the
algebra A(R) generated by the elements of the matrices L^\pm(z) with relations
determined by R. In the category of highest weight representations, there is a
Hopf algebra isomorphism between A(R) and an extension
\overline{U}_q(\widehat{sl}_2)} of Drinfeld's algebra.


title: Twisted Quantum Affine Algebras

abstract:
  We classify the irreducible finite-dimensional representations of the twisted
quantum affine algebras.


title: Topics in hidden symmetries. III

abstract:
  This short paper being devoted to some aspects of the inverse problem of the
representation theory treats several themes, which have their origins in the
researches of F.A.Berezin, D.P.Zhelobenko, V.P.Maslov and his group, in context
of the author's approach to the setting free of hidden symmetries, which is
based on the noncommutative geometry. Relations of the discussed topics to the
general ideologies of the translator algebras in representation theory and of
the dynamical symmetries or the spectrum--generating algebras in mathematical
physics are pointed out. A possibility of quantum field generalizations of the
constructions in terms of vertex (QFT--operator) and Zamolodchikov algebras is
mentioned.


title: Formal GNS Construction and WKB Expansion in Deformation Quantization

abstract:
  In this contribution we review the formal GNS construction developped in a
previous preprint (q-alg/9607019), and formulate the usual WKB-expansion in
flat 2n-dimensional phase space in terms of a GNS construction with a positive
linear functional with support on a projectable Lagrangean submanifold defined
as a graph of an exact one form dS. The main trick is a suitable form of the
star-exponential of S.


title: On the q-analog of homological algebra

abstract:
  This is an attempt to generalize some basic facts of homological algebra to
the case of "complexes" in which the differential satisfies the condition
$d^N=0$ instead of the usual $d^2=0$. Instead of familiar sign factors, the
constructions related to such "N-complexes" involve powers of q where q is a
primitive Nth root of 1. We show that the homology (in a natural sense) of an
N-complex is an $(N-1)$-complex which is $(N-1)$-exact, and the role of the
Euler characteristic is played by the trigonometric sum $\sum q^i \dim(C^i)$.
By q-deforming the de Rham differential we develop a version of the theory of
differential forms which is coordinate-dependent but covariant with respect to
a natural Hopf algebra. In particular, there is a meaningful formalism of
connections with the curvature being an N-form given by the N th power of the
covariant derivative. For $N=3$ the expression for the curvature is very
similar to the Chern-Simons functional. This text was written in 1991.


title:
Braided Geometry and the Inductive Construction of Lie Algebras and
  Quantum Groups

abstract:
  Double-bosonisation associates to a braided group in the category of modules
of a quantum group, a new quantum group. We announce the semiclassical version
of this inductive construction.


title: Weak C* Hopf Symmetry

abstract:
  Weak C* Hopf algebras can act as global symmetries in low-dimensional quantum
field theories, when braid group statistics prevents group symmetries.
Possibilities to construct field algebras with weak C* Hopf symmetry from a
given theory of local observables are discussed.


title:
On inner product in modular tensor categories. II. Inner product on
  conformal blocks and affine inner product identities

abstract:
  This is the second part of the paper (the first part is published in Jour. of
AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every
modular tensor category (MTC) inner products on the spaces of morphisms and
proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i,
U)$ is modular invariant. Also, we have shown that in the case of the MTC
arising from the representations of the quantum group $U_q \sln$ at roots of
unity and $U$ being a symmetric power of the fundamental representation, this
inner product coincides with so-called Macdonald's inner product on symmetric
polynomials.
  In this paper, we apply the same construction to the MTC coming from the
integrable representations of affine Lie algebras. In this case our
construction immediately gives a hermitian form on the spaces of conformal
blocks, and this form is modular invariant (Warning: we cannot prove that it is
positive definite). We show that this form can be rewritten in terms of
asymptotics of KZ equations, and calculate it for $sl_2$, in which case the
formula is a natural affine analogue of Macdonald's inner product identities.
We also formulate as a conjecture similar formula for $sl_n$.


title:
Finite dimensional unitary representations of quantum Anti-de Sitter
  groups at roots of unity

abstract:
  We study irreducible unitary \reps of $U_q(SO(2,1))$ and $U_q(SO(2,3))$ for
$q$ a root of unity, which are finite dimensional. Among others, unitary \reps
corresponding to all classical one-particle representations with integral
weights are found for $q = e^{i \pi /M}$, with $M$ being large enough. In the
"massless" case with spin bigger than or equal to 1 in 4 dimensions, they are
unitarizable only after factoring out a subspace of "pure gauges", as
classically. A truncated associative tensor product describing unitary
many-particle representations is defined for $q = e^{i\pi /M}$.


title: Vertex Operators - From a Toy Model to Lattice Algebras

abstract:
  Within the framework of the discrete Wess-Zumino-Novikov-Witten theory we
analyze the structure of vertex operators on a lattice. In particular, the
lattice analogues of operator product expansions and braid relations are
discussed. As the main physical application, a rigorous construction for the
discrete counterpart g_n of the group valued field g(x) is provided. We study
several automorphisms of the lattice algebras including discretizations of the
evolution in the WZNW model. Our analysis is based on the theory of modular
Hopf algebras and its formulation in terms of universal elements. Algebras of
vertex operators and their structure constants are obtained for the deformed
universal enveloping algebras U_q(G). Throughout the whole paper, the abelian
WZNW model is used as a simple example to illustrate the steps of our
construction.


title:
BC-type interpolation Macdonald polynomials and binomial formula for
  Koornwinder polynomials

abstract:
  We consider 3-parametric polynomials which replace the A-series interpolation
Macdonald polynomials in the BC case. For these polynomials, we prove: an
integral representation, a combinatorial formula, Pieri-type rules, Cauchy
identity, and we also show that they do not satisfy any rational q-difference
equation. We also prove a binomial formula for the 6-parametric Koornwinder
polynomials.


title: Variation on a theme of Selberg integral

abstract:
  In this paper we calculate some Generalized Selberg integrals. The answer is
expressed in terms of $\Gamma$-functions. Integrals of this type serve as
normalization constants or directly via undoing 2-D integrals for determination
of structural constants of operator algebra.


title: On boson algebras as Hopf algebras

abstract:
  Certain types of generalized undeformed and deformed boson algebras which
admit a Hopf algebra structure are introduced, together with their Fock-type
representations and their corresponding $R$-matrices. It is also shown that a
class of generalized Heisenberg algebras including those algebras including
those underlying physical models such as that of Calogero-Sutherland, is
isomorphic with one of the types of boson algebra proposed, and can be
formulated as a Hopf algebra.


title:
Induced Hopf stucture and irreducible representations of an elliptic
  $U_{q,p}(sl(2))$ via a nonlinear map

abstract:
  Shiraishi's two parameter generalization of $U_{q}(sl(2))$ to
$U_{q,p}(sl(2))$ involving an elliptic function is considered. The generators
are mapped non-linearly on those of $U_{q}(sl(2))$. This gives directly the
irreducible representations and an induced Hopf structure. This is one
particular example of the scope of a class of non-linear maps introduced by us
recently.


title: A Generalization of $U_h(sl(2))$ via Jacobian Elliptic Function

abstract:
  A two-parametric generalization of the Jordanian deformation $U_h (sl(2))$ of
$sl(2)$ is presented. This involves Jacobian elliptic functions. In our
deformation $U_{(h,k)}(sl(2))$, for $k^2=1$ one gets back $U_h(sl(2))$. The
constuction is presented via a nonlinear map on $sl(2)$. This invertible map
directly furnishes the highest weight irreducible representations of
$U_{(h,k)}(sl(2))$. This map also provides two distinct induced Hopf stuctures,
which are exhibited. One is induced by the classical $sl(2)$ and the other by
the distinct one of $U_h(sl(2))$. Automorphisms related to the two periods of
the elliptic functions involved are constructed. Translations of one generator
by half and quarter periods lead to interesting results in this context.
Possibilities of applications are discussed briefly.


title: Knot Theory of Coxeter type B and its physical applications

abstract:
  Braid groups may be defined for every Coxeter diagram. Artin's braid group is
of type A. Analogs of Temperley-Lieb, Hecke and Birman-Wenzl algebras exist for
B-type. Our general hypothethis is that the braid group of B-type replaces
Artin's braid group in most physical applications if the model is equipped with
a nontrivial boundary. Solutions of a Potts model with a boundary and the
reflection equation illustrate this principle. Braided tensor categories of
B-type and dually Coxeter-B braided Hopf algebras are introduced. The
occurrence of such categories in QFT on a half plane is discussed.


title: Actions of Pointed Hopf Algebras

abstract:
  Action of finite-dimensional Hopf algebra $H$ on commutative $k-$algebra $A$
is considered. As a generalization of the well-known fact for finite groups S.
Montgomery raised a problem in 1993 whether $A$ is integral over subalgebra of
invariants $A^H$. Recently some new results were obtained. Using the properties
of coradical filtration of pointed Hopf algebras we verified the truth of the
hypothesis in tree different cases: 1) Hopf algebra $H$ is commutative; 2) char
$k = p > 0$; 3) $A$ is integral domain. In spite of numerous partial positive
results it turned out that hypothesis of S. Montgomery isn't true in general.
The counteraxamples were built for series of pointed Hopf algebras $A_N, N \ge
2$.


title: Factorizable D-modules

abstract:
  A braided tensor category $FM_{\kappa}$ of `factorizable D-modules' over
configuration spaces is introduced, analogous to the category $FS_q$ of
factorizable sheaves from q-alg/9604001. This category is equivalent to the
category of finite dimensional representations of a complex semisimple Lie
algebra $\frak{g}$, with the Drinfeld's Knizhnik-Zamolodchikov tensor product.
This description, together with the result of op.cit., gives a new,
"Riemann-Hilbert" proof of the Drinfeld's theorem establishing an equivalence
of the above tensor category with the category of finite dimensional
$U_q\frak{g}$-modules ($q=\exp(2\pi i/kappa)$, $\kappa$ irrational).


title: Topics in hidden symmetries. IV

abstract:
  This note being devoted to some aspects of the inverse problem of
representation theory explicates the links between researches on the Sklyanin
algebras and the author's (based on the noncommutative geometry) approach to
the setting free of hidden symmetries in terms of "the quantization of
constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is
constructed. It may be considered as a result of the quantization of constants
in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner
algebra for the Sklyanin algebra is an example of the noncommutative weighted
shift operator algebras (NWSO-algebras), which generalize the mho-algebras
introduced by the author earlier. If the Sklyanin algebra is interpreted as an
algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded
as an enlargement of the Sklyanin algebra by operators of the anomalous
spin-spin interaction (of tensor type).


title: Lickorish Invariant And Quantum OSP(1|2)

abstract:
  Lickorish's method for constructing topological invariants of 3 - manifolds
is generalized to the quantum supergroup setting. An invariant is obtained by
applying this method to the Kauffman polynomial arising from the vector
representation of Uq(osp(1|2)). A transparent proof is also given showing that
this invariant is equivalent to the Uq(osp(1|2)) invariant obtained in an
earlier publication.


title: Minimal Uncertainty States For Quantum Groups

abstract:
  The problem of how to obtain quasi-classical states for quantum groups is
examined. A measure of quantum indeterminacy is proposed, which involves
expectation values of some natural quantum group operators. It is shown that
within any finite dimensional irreducible representation, the highest weight
vector and those unitarily related to it are the quasi-classical states.


title:
Deformation quantization of compact K\"ahler manifolds via
  Berezin-Toeplitz operators

abstract:
  This talk reports on results on the deformation quantization (star products)
and on approximative operator representations for quantizable compact K"ahler
manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic
quantum line bundle the Berezin-Toeplitz operator associated to a
differentiable function on the manifold is the operator defined by multiplying
global holomorphic sections of the line bundle with this function and
projecting the differentiable section back to the subspace of holomorphic
sections. The results were obtained in (respectively based on) joint work with
M. Bordemann and E. Meinrenken. (Talk at the XXI Int. Coll. on Group
Theoretical Methods in Physics, 15-20 July, Goslar, Germany)

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


title: C^*-algebraic quantum groups arising from algebraic quantum groups

abstract:
  We associate to an algebraic quantum group a C^*-algebraic quantum group and
prove that this C^*-algebraic quantum group satisfies an upcoming definition of
Masuda, Nakagami & Woronowicz.


title: Structure And Representations Of The Quantum General Linear Supergroup

abstract:
  The structure and representations of the quantum general linear supergroup
GLq(m|n) are studied systematically by investigating the Hopf superalgebra Gq
of its representative functions. Gq is factorized into $Gq^{\pi} Gq^{\bar\pi}$,
and a Peter - Weyl basis is constructed for each factor. Parabolic induction
for the quantum supergroup is developed. The underlying geometry of induced
representations is discussed, and an analog of Frobenius reciprocity is
obtained. A quantum Borel - Weil theorem is proven for the covariant and
contravariant tensorial irreps, and explicit realizations are given for classes
of tensorial irreps in terms of sections of quantum super vector bundles over
quantum projective superspaces.


title:
The Hopf algebra isomorphism between kappa-Poincare algebra in the case
  g^{00}=0 and "null plane" quantum Poincare algebra

abstract:
  The Hopf algebra isomorphism between kappa-Poincare algebra defined by
P.Kosinski and P.Maslanka in the case g^{00}=0 and "null plane" quantum
Poincare algebra by A.Ballesteros, F.J.Herranz and M.A.del Olmo is defined.


title: Enhancing an R-matrix

abstract:
  In order to construct a representation of the tangle category one needs an
enhanced R-matrix. In this paper we define a sufficient and necessary condition
for enhancement that can be checked easily for any R-matrix. If the R-matrix
can be enhanced, we also show how to construct the additional data that define
the enhancement. As a direct consequence we find a sufficient condition for the
construction of a knot invariant.


title: Calogero-Moser pairs and the Airy and Bessel bispectral involutions

abstract:
  Explicit formulae are given for the Airy and Bessel bispectral involutions,
in terms of Calogero-Moser pairs. Hamiltonian structure of the motion of the
poles of the operators is discussed.


title: Quasi Hopf Deformations of Quantum Groups

abstract:
  The search for elliptic quantum groups leads to a modified quantum
Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf
algebras. This paper calculates deformations of standard quantum groups (with
or without spectral parameter) in the category of quasi-Hopf algebras. An
earlier investigation of the deformations of quantum groups, in the category of
Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra
deformations exist only under some restrictions on the parameters. In
particular, affine Kac-Moody algebras are more rigid than their loop algebra
quotients; and only the latter (in the case of sl(n)) can be deformed to
elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts
this restriction. The full elliptic quantum groups (with central extension)
associated with $sl(n)$ are thus quasi-Hopf algebras. The universal R-matrices
satisfy a modified Yang-Baxter relation and are calculated more or less
explicitly. The modified classical Yang-Baxter relation is obtained, and the
elliptic solutions are worked out explicitly. The same method is used to
construct the Universal R-matrices associated with Felder's quantization of the
Knizhnik-Zamolodchikov-Bernard equation, to throw some light on the quasi Hopf
structure of conformal field theory and (perhaps) the Calogero-Moser models.


title: Bicharacters, braids and Jacobi identity

abstract:
  For an abelian group G we consider braiding in a category of G-graded modules
$M^{kG}$ given by a bicharacter \chi on G. For $(G,\chi)$-bialgebra A in
$M^{kG}$ an analog of Lie bracket is defined. This bracket is determined by a
linear map $E\in\End(A)$ and n-ary operations $\Omega^{n}_{E}$ on A. Our result
states that if $E(1)=0,E^{2}=0$ and $\Omega^{3}_{E}=0$ then a braided Jacobi
identity holds and the linear map E is a braided derivation of a braided Lie
algebra.


title: Quantum toroidal algebras and their vertex representations

abstract:
  We change the definition of the vertex representations. As a result the
vertex representations has one parameter.


title: Boson representations, non-standard quantum algebras and contractions

abstract:
  A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the
non-standard quantum deformation of $sl(2,\R)$ which directly provides its
infinite and finite dimensional irreducible representations. Tensor product
decompositions are worked out for some examples. Relations between contraction
methods and boson realizations are also explored in several contexts. So, a
class of two-boson representations for the non-standard deformation of
$sl(2,\R)$ is introduced and contracted to the non-standard quantum (1+1)
Poincar\'e representations. Likewise, a quantum extended Hopf $sl(2,\R)$
algebra is constructed and the Jordanian $q$-oscillator algebra representations
are obtained from it by means of another contraction procedure.


title:
Combinatorial Structure of Finite Dimensional Representations of
  Yangians: the Simply-Laced Case

abstract:
  We compute the decomposition of representations of Yangians into g-modules
for simply-laced Lie algebras g. The decomposition has an interesting
combinatorial tree structure. Results depend on a conjecture of Kirillov and
Reshetikhin.


title: Tensor ideals in the category of tilting modules

abstract:
  We study the tensor category $\cQ$ of tilting modules over a quantum group
$U_q$ with divided powers. The set $X_+$ of dominant weights is a union of
closed alcoves $\oC_w$ numbered by the elements $w\in W^f$ of a certain subset
of affine Weyl group $W$. G.Lusztig and N.Xi defined a partition of $W^f$ into
canonical right cells and the right order $\le_R$ on the set of cells. For a
cell $A\subset W^f$ we consider a full subcategory $\cQ_{<A}$ formed by direct
sums of tilting modules $Q(\lambda)$ with highest weights $\lambda \in
\bigcup_{w\in B<_RA} \oC_w$. We prove that $\cQ_{<A}$ is a tensor ideal in
$\cQ$, generalizing H.Andersen's Theorem about the ideal of negligible modules
which in our notations is nothing else then $\cQ_{<\{ e\}}$. The proof is an
application of a recent result by W.Soergel who has computed the characters of
tilting modules.


title: Quantum Deformation of the W_N Algebra

abstract:
  We review the W_N algebra and its quantum deformation, based on free field
realizations. The (quantum deformed) W_N algebra is defined through the
(quantum deformed) Miura transformation, and its singular vectors realize the
Jack (Macdonald) polynomials. (Talk at the Nankai-CRM joint meeting on
``Extended and Quantum Algebras and their Applications to Physics'', Tianjin,
China, August 19-24, 1996.)


title: On Lie Algebras in Braided Categories

abstract:
  The set of primitive elements of a Hopf algebra in the braided category of
group graded vector spaces (with a commutative group) carry the structure of a
generalized Lie algebra. In particular the graded derivations of an associative
algebra carry this Lie algebra structure. The Lie multiplications consist of
certain n-ary partially defined multiplications satisfying generalized
antisymmetry and Jacobi identities. This generalizes the concept of Lie super
algebras and Lie color algebras.
  We show that universal enveloping algebras in the braided category exist.
They are (braided) Hopf algebras. This explains many constructions of
noncommutative noncocommutative Hopf algebras in the literature.


title: Non-Symmetric Jack Polynomials and Integral Kernels

abstract:
  We investigate some properties of non-symmetric Jack, Hermite and Laguerre
polynomials which occur as the polynomial part of the eigenfunctions for
certain Calogero-Sutherland models with exchange terms. For the non-symmetric
Jack polynomials, the constant term normalization ${\cal N}_\eta$ is evaluated
using recurrence relations, and ${\cal N}_\eta$ is related to the norm for the
non-symmetric analogue of the power-sum inner product. Our results for the
non-symmetric Hermite and Laguerre polynomials allow the explicit determination
of the integral kernels which occur in Dunkl's theory of integral transforms
based on reflection groups of type $A$ and $B$, and enable many analogues of
properties of the classical Fourier, Laplace and Hankel transforms to be
derived. The kernels are given as generalized hypergeometric functions based on
non-symmetric Jack polynomials. Central to our calculations is the construction
of operators $\widehat{\Phi}$ and $\widehat{\Psi}$, which act as lowering-type
operators for the non-symmetric Jack polynomials of argument $x$ and $x^2$
respectively, and are the counterpart to the raising-type operator $\Phi$
introduced recently by Knop and Sahi.


title: Quasi-Continuous Symmetries of Non-Lie Type

abstract:
  We introduce a smooth mapping of some discrete space-time symmetries into
quasi-continuous ones. Such transformations are related with q-deformations of
the dilations of the Euclidean space and with the non-commutative space. We
work out two examples of Hamiltonian invariance under such symmetries. The
Schrodinger equation for a free particle is investigated in such a
non-commutative plane and a connection with anyonic statistics is found.


title: Nullification Writhe and Chirality of Alternating Links

abstract:
  In this paper, we show how to split the writhe of reduced projections of
oriented alternating links into two parts, called nullification writhe, or wx,
and remaining writhe, or wy, such that the sum of these quantities equals the
writhe w, and each quantity remains an invariant of isotopy. The chirality of
oriented alternating links can be detected by a non-zero wx or wy, which
constitutes an improvement compared to the detection of chirality by a non-zero
w. An interesting corollary is that all oriented alternating links with an even
number of components are chiral, a result that also follows from properties of
the Conway polynomial.


title: Generalized KdV Equation for Fluid Dynamics and Quantum Algebras

abstract:
  We generalize the non-linear one-dimensional equation of a fluid layer for
any depth and length as an infinite order differential equation for the steady
waves. This equation can be written as a q-differential one, with its general
solution written as a power series expansion with coefficients satisfying a
nonlinear recurrence relation. In the limit of long and shallow water (shallow
channels) we reobtain the well known Korteweg-de-Vries equation together with
its single-soliton solution.


title:
Complex su_q(2) Dynamical Symmetry, Limiting Cases and 1-Dimensional
  Potential Realisation

abstract:
  Using a complex deformation q=exp(is) of su(2) we obtain extensions of the
finite-dimensional representations towards the infinite-dimensional ones. A
generalised q-deformation of su(2), as a Hopf algebra is introduced. We present
the corresponding Schrodinger picture, by using a differential realisation, and
a large class of potentials is obtained.A connection between the unirreps with
q a root of unity and the comensurability of the potentials is investigated.
The smooth transition between su(2) and su(1,1) , through E(1) is obtained at
different levels: of the unirreps, of the topology, of the commutators and of
the potentials in the corresponding Schrodinger equation.


title: Hopf algebra extension of a Zamolochikov algebra and its double

abstract:
  The particles with a scattering matrix R(x) are defined as operators
$\Phi_i(z)$ satisfying the relation $ R_{i,j}^{j',i'}(x_1/x_2)
\Phi_{i'}(x_1)\Phi_{j'}(x_2)= \Phi_i(x_2)\Phi_j(x_1)$. The algebra generated by
those operators is called a Zamolochikov algebra. We construct a new Hopf
algebra by adding half of the FRTS construction of a quantum affine algebra
with this R(x). Then we double it to obtain a new Hopf algebra such that the
full FRTS construction of a quantum affine algebra is a Hopf subalgebra inside.
Drinfeld realization of quantum affine algebras is included as an example. This
is a further generalization of the constructions in q-alg/9608002.


title:
Quantum current operators (III): Commutative quantum current operators,
  semi-infinite construction and functional models

abstract:
  We construct a commutative current operator $\bar x^+(z)$ inside
$U_q(\hat{\frak sl}(2))$. With this operator and the condition of quantum
integrability on the quantum current of $U_q(\hat{\frak sl}(2))$, we derive the
quantization of the semi-infinite construction of integrable modules of
$\hat{\frak sl}(2)$ with the current operator $e(z)$ of $\hat{\frak sl}(2)$.
The quantization of the functional models for $\hat{\frak sl}(2)$ are also
given.


title: Vertex operator algebras and associative algebras

abstract:
  Let V be a vertex operator algebra. We construct a sequence of associative
algebras A_n(V) (n=0,1,2,...) such that A_{n}(V) is a quotient of A_{n+1}(V)
and a pair of functors between the category of A_n(V)-modules which are not
A_{n-1}(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 rational if and only if all A_n(V) are finite-dimensional semisimple
algebras.


title: Deformations of (Bi) tensor Categories

abstract:
  We define the cohomology and formal deformation theories for algebra and
bialgebra categories. We suggest some approaches to finding nontrivial
deformations of the categories associated to the quantum groups by the work of
Lusztig.


title: Drinfel'd Twists and Algebraic Bethe Ansatz

abstract:
  We study representation theory of Drinfel'd twists, in terms of what we call
F matrices, associated to finite dimensional irreducible modules of quantum
affine algebras, and which factorize the corresponding (unitary) R matrices. We
construct explicitly such factorizing F matrices for irreducible tensor
products of the fundamental representations of the quantum affine algebra sl2
and its associated Yangian. We then apply these constructions to the XXX and
XXZ quantum spins chains of finite length in the framework of the Algebraic
Bethe Ansatz.

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


title: Crossed Product Structure of Quantum Euclidean Groups

abstract:
  It is shown that quantum Euclidean groups $E_q(2)$, $E_\kappa(2)$ and
$E_\kappa(3)$ have the structure of generalised crossed products.


title:
Integral soluitons of q-difference equations of the hypergeometric type
  with |q|=1

abstract:
  Two integral solutions of q-difference equations of the hypergeometric type
with |q|=1 are constructed by using the double sine function. One is an
integral of the Barnes type and the other is of the Euler type.


title:
Holomorphic deformation of Hopf algebras and applications to quantum
  groups

abstract:
  In this article we propose a new and so-called holomorphic deformation scheme
for locally convex algebras and Hopf algebras. Essentially we regard converging
power series expansion of a deformed product on a locally convex algebra, thus
giving the means to actually insert complex values for the deformation
parameter. Moreover we establish a topological duality theory for locally
convex Hopf algebras. Examples coming from the theory of quantum groups are
reconsidered within our holomorphic deformation scheme and topological duality
theory. It is shown that all the standard quantum groups comprise holomorphic
deformations. Furthermore we show that quantizing the function algebra of a
(Poisson) Lie group and quantizing its universal enveloping algebra are
topologically dual procedures indeed. Thus holomorphic deformation theory seems
to be the appropriate language in which to describe quantum groups as deformed
Lie groups or Lie algebras.


title: Bott - Borel - Weil Construction For Quantum Supergroup Uq(gl(m|n))

abstract:
  The finite dimensional irreducible representations of the quantum supergroup
$U_q(gl(m|n))$ are constructed geometrically using techniques from the Bott -
Borel - Weil theory and vector coherent states.


title: Quommutator deformations of osp(2,2)

abstract:
  We analyse the Witten-Woronowicz's type deformations of the Lie superalgebra
osp(2,2) and obtain a deformation parametrized by three independent parameters.
For some of these algebras, finite dimensional representations are formulated
in terms of a finite difference operator, providing operators that are relevant
for the classification of quasi exactly solvable, finite difference equations.
Similar representations are also pointed out for the Lie superalgebra osp(1,2).


title: Paths, Demazure Crystals and Symmetric Functions

abstract:
  We review the path realization of Demazure crystals and discuss Demazure
characters in the light of symmetric functions.


title: Cohomological Properties of Differential Calculi on Hopf Algebras

abstract:
  In this report we give an intrinsic treatment of the results we developed in
a previous work connecting the differential calculi on Hopf algebras to the
Drinfeld double. In the first place we recover that bicovariant bimodules are
in one to one correspondence with the Drinfeld double representations; we then
introduce a Hochschild cohomology of the algebra of functions and discuss the
main result stating that each differential calculus is associated to a
1-cocycle satisfying an additional invariance condition with respect to a
natural action. Defining a Hochschild cohomology of the double, the above
invariance becomes a condition with respect to the enveloping algebra component
of the double that must be added to the 1-cocycle relation.


title: Symmetrizable Quantum Affine Superalgebras And Their Representations

abstract:
  Aspects of the algebraic structure and representation theory of the quantum
affine superalgebras with symmetrizable Cartan matrices are studied. The
irreducible integrable highest weight representations are classified, and shown
to be deformations of their classical counterparts. It is also shown that Jimbo
type quantum affine superalgebras can be obtained by deforming universal
enveloping algebras of ordinary (i.e., non-graded) affine algebras supplemented
by certain parity operators.


title: Central Extensions of the Quasi-orthogonal Lie algebras

abstract:
  We determine the central extensions of a whole family of Lie algebras,
obtained by the method of graded contractions from so(N+1), N arbitrary. All
the inhomogeneous orthogonal and pseudo-orthogonal algebras are members of this
family, as well as a large number of other non-semisimple algebras, all of
which have at least a semidirect structure (in some cases two or more). The
dimensions of their second cohomology groups H^2(G,R) and the explicit
expression of their central extensions are given.


title:
Graded contractions and bicrossproduct structure of deformed
  inhomogeneous algebras

abstract:
  A family of deformed Hopf algebras corresponding to the classical maximal
isometry algebras of zero-curvature N-dimensional spaces (the inhomogeneous
algebras iso(p,q), p+q=N, as well as some of their contractions) are shown to
have a bicrossproduct structure. This is done for both the algebra and, in a
low-dimensional example, for the (dual) group aspects of the deformation.


title: On Lie Algebras in the Category of Yetter-Drinfeld Modules

abstract:
  The category of Yetter-Drinfeld modules over a Hopf algebra (with bijektive
antipode over a field) is a braided monoidal category. Given a Hopf algebra in
this category then the primitive elements of this Hopf algebra do not form an
ordinary Lie algebra anymore.
  We introduce the notion of a (generalized) Lie algebra in the category of
Yetter-Drinfeld modules such that the set of primitive elements of a Hopf
algebra is a Lie algebra in this sense. It has n-ary partially defined Lie
multiplications on certain symmetric submodules of n- fold tensor products.
They satisfy antisymmetry and Jacobi identities. Also the Yetter-Drinfeld
module of derivations of an associative algebra in the category of Yetter-
Drinfeld modules is a Lie algebra. Furthermore for each Lie algebra in the
category of Yetter-Drinfeld modules there is a universal enveloping algebra
which turns out to be a (braided) Hopf algebra in this category.


title: Fermion Representation of Quantum Group

abstract:
  The spinor representation of the quantum group $U_q(su(N))$ is given in terms
of a set of fermion creation and annihilation operators. It is shown that the
$q$-fermion operators introduced earlier can be identifi ed with the
conventional fermion operators. Algebra homomorphisms mapping the fermion
operators to their t ensor products are discussed. The relation of the
coproduct of the quantum group to the above algebra homo morphisms is obtained.


title:
Shifted Schur functions II. Binomial formula for characters of classical
  groups and applications

abstract:
  Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n),
let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of
G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type
expansion for finite-dimensional characters of G (binomial formula) and use it
to specify a distinguished linear basis in Z(g). The eigenvalues of the basis
elements in highest weight g-modules are certain shifted (or factorial) analogs
of Schur functions. We also study an associated homogeneous basis in I(g), the
subalgebra of G-invariants in the symmetric algebra S(g). Finally, we show that
the both bases are related by a G-equivariant linear isomorphism \sigma:
I(g)\to Z(g), called the special symmetrization.


title: Hidden symmetries and categorical representation theory

abstract:
  The interrelations between the inverse problems of the representation theory
and the categorical representation theory are discussed.


title:
The Lie Algebraic Structure of Differential Operators Admitting
  Invariant Spaces of Polynomials

abstract:
  We prove that the scalar and $2\times 2$ matrix differential operators which
preserve the simplest scalar and vector-valued polynomial modules in two
variables have a fundamental Lie algebraic structure. Our approach is based on
a general graphical method which does not require the modules to be irreducible
under the action of the corresponding Lie (super)algebra. This method can be
generalized to modules of polynomials in an arbitrary number of variables. We
give generic examples of partially solvable differential operators which are
not Lie algebraic. We show that certain vector-valued modules give rise to new
realizations of finite-dimensional Lie superalgebras by first-order
differential operators.


title: Combined (q,h)-Deformation as a Nonlinear Map on $U_q(sl(2))$

abstract:
  The generators $(J_{\pm}, J_0)$ of the algebra $U_q(sl(2))$ is our starting
point. An invertible nonlinear map involving, apart from q, a second arbitrary
complex parameter h, defines a triplet $({\hat X},{\hat Y},{\hat H})$. The
latter set forms a closed algebra under commutation relations. The nonlinear
algebra $U_{q,h}(sl(2))$, thus generated, has two different limits. For $q \to
1$, the Jordanian h-deformation $U_{h}(sl(2))$ is obtained. For $h \to 0$, the
q-deformed algebra $U_{q}(sl(2))$ is reproduced. From the nonlinear map, the
irreducible representations of the doubly-deformed algebra $U_{q,h}(sl(2))$ may
be directly and explicitly obtained form the known representations of the
algebra $U_q(sl(2))$. Here we consider only generic values of q.


title: Discrete spectral triples and their symmetries

abstract:
  We classify 0-dimensional spectral triples over complex and real algebras and
provide some general statements about their differential structure. We
investigate also whether such spectral triples admit a symmetry arising from
the Hopf algebra structure of the finite algebra. We discuss examples of
commutative algebras and groups algebras.


title: Differential calculi on noncommutative bundles

abstract:
  We introduce a category of noncommutative bundles. To establish geometry in
this category we construct suitable noncommutative differential calculi on
these bundles and study their basic properties. Furthermore we define the
notion of a connection with respect to a differential calculus and consider
questions of existence and uniqueness. At the end these constructions are
applied to basic examples of noncommutative bundles over a coquasitriangular
Hopf algebra.


title: Poisson Algebra of Differential Forms

abstract:
  We give a natural definition of a Poisson Differential Algebra. Consistence
conditions are formulated in geometrical terms. It is found that one can often
locally put the Poisson structure on differential calculus in a simple
canonical form by a coordinate transformation. This is in analogy with the
standard Darboux's theorem for symplectic geometry. For certain cases there
exists a realization of the exterior derivative through a certain canonical
one-form. All the above are carried out similarly for the case of a complex
Poisson Differential Algebra. The case of one complex dimension is treated in
details and interesting features are noted. A conclusion is made in the last
section.


title: R-matrix Quantization of the Elliptic Ruijsenaars--Schneider model

abstract:
  It is shown that the classical L-operator algebra of the elliptic
Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of
functions on the cotangent bundle over the centrally extended current group in
two dimensions. It is governed by two dynamical r and $\bar{r}$-matrices
satisfying a closed system of equations. The corresponding quantum R and
$\overline{R}$-matrices are found as solutions to quantum analogs of these
equations. We present the quantum L-operator algebra and show that the system
of equations on R and $\overline{R}$ arises as the compatibility condition for
this algebra. It turns out that the R-matrix is twist-equivalent to the Felder
elliptic R^F-matrix with $\overline{R}$ playing the role of the twist. The
simplest representation of the quantum L-operator algebra corresponding to the
elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum
L-operator algebra to the fundamental relation RLL=LLR with Belavin's elliptic
R matrix is established. As a byproduct of our construction, we find a new
N-parameter elliptic solution to the classical Yang-Baxter equation.


title: Twisted Wess-Zumino-Witten models on elliptic curves

abstract:
  Investigated is a variant of the Wess-Zumino-Witten model called a twisted
WZW model, which is associated to a certain Lie group bundle on a family of
elliptic curves. The Lie group bundle is a non-trivial bundle with flat
connection and related to the classical elliptic r-matrix. (The usual
(non-twisted) WZW model is associated to a trivial group bundle with trivial
connection on a family of compact Riemann surfaces and a family of its
principal bundles.) The twisted WZW model on a fixed elliptic curve at the
critical level describes the XYZ Gaudin model. The elliptic
Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix
appear as flat connections on the sheaves of conformal blocks in the twisted
WZW model.


title: Topological Invariants For Lens Spaces And Exceptional Quantum Groups

abstract:
  The Reshetikhin - Turaev invariants arising from the quantum groups
associated with the exceptional Lie algebras $G_2$, $F_4$ and $E_8$ at odd
roots of unity are constructed and explicitly computed for all the lens spaces.


title: Double-loop algebras and the Fock space

abstract:
  The fermionic Fock space admits two different actions of the quantized
enveloping algebra of $\hat\sln$> The first one is a q-deformation of the
well-known level-one representation of the affine Lie algebra and the second
one is a new level-zero action arising from solvable lattices models. In this
paper we prove that these two constructions can be glued together to get a
representation of a new object: the toridal quantum group. This representation
involves some difference operatorators. It is due to the fact that the
specialization at q=1 of the toroidal algebra is isomorphic to the enveloping
algebra of the universal central extension of a current algebra over a quantum
torus.


title:
q-differential operator representation of the quantum superalgebra
  Uq(sl(M+1|N+1))

abstract:
  A representation of the quantum superalgebra Uq(sl(M+1|N+1)) is constructed
based on the q-differential operators acting on the coherent states
parameterized by coordinates. These coordinates correspond to the local ones of
the flag manifold. This realization provides us with a guide to construct the
free field realization for the quantum affine superalgebra Uq^(sl(M+1|N+1)) at
arbitrary level.


title: Quantum Moduli Spaces of Flat Connections

abstract:
  Using the formalism of discrete quantum group gauge theory, one can construct
the quantum algebras of observables for the Hamiltonian Chern-Simons model. The
resulting moduli algebras provide quantizations of the algebra of functions on
the moduli spaces of flat connections on a punctured 2-dimensional surface. In
this note we describe some features of these moduli algebras with special
emphasis on the natural action of mapping class groups. This leads, in
particular, to a closed formula for representations of the mapping class groups
on conformal blocks.


title: Quantum Geometry and the Planck Scale

abstract:
  We consider some general aspects of the new noncommutative or quantum
geometry coming out of the theory of quantum groups, in connection with Planck
scale physics. A generalisation of Fourier or wave-particle duality on curved
spaces emerges. Another feature is the need for particles with fractional or
braid statistics. The conformal group also has a special role.


title: Quantum Double for QuasiHopf Algebras

abstract:
  We introduce a quantum double quasitriangular quasi-Hopf algebra $D(H)$
associated to any quasi-Hopf algebra $H$. The algebra structure is a cocycle
double cross product. We use categorical reconstruction methods. As an example,
we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the
quantum double $D^\phi(G)$ associated to a finite group $G$ and group 3-cocycle
$\phi$.


title: Exact n-spinon dynamic structure factor of the Heisenberg model

abstract:
  Using the spinon picture we derive an integral representation for the exact
n-spinon dynamic structure factor of the spin-1/2 Heisenberg model.


title:
A Hopf algebra isomorphism between two realizations of the quantum
  affine algebra $U_q(\widehat{gl(2)})$

abstract:
  We consider the algebra isomorphism found by Frenkel and Ding between the RLL
and the Drinfeld realizations of $U_q(\widehat{gl(2)})$. After we note that
this is not a Hopf algebra isomorphism, we prove that there is a unique Hopf
algebra structure for the Drinfeld realization so that this isomorphism becomes
a Hopf algebra isomorphism. Though more complicated, this Hopf algebra
structure is also closed, just as the one found previously by Drinfeld.


title: Quantum Schubert polynomials and quantum Schur functions

abstract:
  We introduce the quantum multi-Schur functions, quantum factorial Schur
functions and quantum Macdonald polynomials. We prove that for restricted
vexillary permutations the quantum double Schubert polynomial coincides with
some quantum multi-Schur function and prove a quantum analog of the
Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert
polynomials in the case of Grassmannian permutations. We prove, also, an analog
of the Billey-Jockusch-Stanley formula for quantum Schubert polynomials.
Finally we formulate two conjectures about the structure of quantum double and
quantum Schubert polynomials for 321-avoiding permutations.


title: Computer Programs for Knot Tabulation

abstract:
  While the problem of knot classification is far from solved, it is possible
to create computer programs that can be used to tabulate knots up to a desired
degree of complexity. Here we discuss the main ideas on which such programs can
be based. We also present the actual results obtained after running a computer
program on which knots are denoted through regular projections.


title: Supernomial coefficients, polynomial identities and $q$-series

abstract:
  $q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N
(1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion
relations, symmetries and limiting theorems of the ``$q$-supernomial
coefficients'' are derived, and a combinatorial interpretation using
generalized Durfee dissection partitions is given. Polynomial identities of
boson-fermion-type, based on the continued fraction expansion of $p/k$ and
involving the $q$-supernomial coefficients, are proven. These include
polynomial analogues of the Andrews-Gordon identities. Our identities unify and
extend many of the known boson-fermion identities for one-dimensional
configuration sums of solvable lattice models, by introducing multiple
finitization parameters.


title:
Factorization of differential operators, quasideterminants, and
  nonabelian Toda field equations

abstract:
  We integrate nonabelian Toda field equations for root systems of types A, B,
C, for functions with values in any associative algebra. The solution is
expressed via quasideterminants. In the appendix we review some results
concerning nonabelian soliton equations.


title: Heisenberg doubles and derived categories

abstract:
  Let A be an abelian category of finite type and homological dimension 1. Then
by results of Green R(A), the extended Hall-Ringel algebra of A, has a natural
Hopf algebra structure. We consider its Heisenberg double Heis(A) and study its
relation with D(A), the derived category of A. We show that Heis(A) can be
viewed as a "Hall algebra" of D^{0,1}(A), the subcategory of complexes situated
in degrees 0 and 1, in the following sense: if B is the heart of a t-structure
on D(A) lying in D^{0,1}(A), then R(B) is naturally a subalgebra in Heis(A).
Further, we define a new algebra L(A) called the lattice algebra of A, obtained
by taking infinitely many copies of R(A), one for each site of an infinite
1-dimensional lattice and imposing Heisenberg double-type relations between
copies at adjacent sites and oscillator-type relations between copies at
non-adjacent sites. This algebra serves as the "Hall algebra" of the full
derived category D(A) in the following sense: any derived equivalence
D(A)-->D(B) induces an isomorphism of lattice algebras L(A)-->L(B).


title: Quantum function algebras as quantum enveloping algebras

abstract:
  Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of
$ F_q[SL(n+1)] $, along with a presentation by generators and relations, and
prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $
{\mathfrak{h}} $ is the Lie bialgebra of the Poisson Lie group $ H $ dual of $
SL(n+1) $; moreover, we explain the relation with [loc. cit.]. In sight of
this, we prove two PBW-like theorems for $ F_q[SL(n+1)] $, both related to the
classical PBW theorem for $ U({\mathfrak{h}}) $.


title: Super Yangian Double $DY( gl(1|1))$ and Its Gauss Decomposition

abstract:
  We extend Yangian double to super (or graded) case and give its Drinfel'd
generators realization by Gauss decomposition.


title:
A new construction of the moonshine vertex operator algebra over the
  real number field

abstract:
  We give a new construction of the moonshine VOA V^{\natural} over the real
number field. We proved that V^{\natural} has a positive definite invariant
bilinear form and its full automorphism group is the Monster simple group. We
also construct an infinite series of meromorphic VOAs whose full automorphism
groups are finite. We calculate the trace form on V^{\natural} for some element
of the Monster.


title: Chromatic weight systems and the corresponding knot invariants

abstract:
  This paper contains a proof that chromatic weight systems, introduced by
Chmutov, Duzhin and Lando, can be expressed in terms of weight systems
associated with direct sums of the Lie algebras gl_n and so_n. As a consequence
the Vassiliev invariants of knots corresponding to the chromatic weight systems
distinguish exactly the same knots as a one variable specialisation Y of the
Homfly and Kauffman polynomial.


title: hbar-(Yangian) Deformation of Miura Map and Virasoro Algebra

abstract:
  An hbar-deformed Virasoro Poisson algebra is obtained using the Wakimoto
realization of the Sugawara operator for the Yangian double DY_\hbar(sl_2)_c at
the critical level c=-2.


title: Free coherent states and $p$-adic numbers

abstract:
  Free coherent states for a system with two degrees of freedom is defined.
Existence of the homeomorphism of the ring of integer 2-adic numbers to the set
of coherent states corresponding to an eigenvalue of the operator of
annihilation is proved. It is shown that the metric of free Fock space induces
the 2-adic topology on the set of coherent states.


title:
The hidden symmetry algebras of a class of quasi-exactly solvable multi
  dimensional operators

abstract:
  Let $P(N,V)$ denote the vector space of polynomials of maximal degree less
than or equal to $N$ in $V$ independent variables. This space is preserved by
the enveloping algebra generated by a set of linear, differential operators
representing the Lie algebra $gl(V+1)$. We establish the counterpart of this
property for the vector space $P(M,V) \oplus P(N,V)$ for any values of the
integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$
generate an abstract superalgebra (non linear if $\Delta=\mid M-N\mid\geq 2$).
A family of algebras is also constructed, extending this particular algebra by
$\Delta -1$ arbitrary complex parameters.


title: Quantizing Poisson Manifolds

abstract:
  This paper extends Kontsevich's ideas on quantizing Poisson manifolds. A new
differential is added to the Hodge decomposition of the Hochschild complex, so
that it becomes a bicomplex, even more similar to the classical Hodge theory
for complex manifolds.


title: Quantization of the Algebra of Chord Diagrams

abstract:
  In this paper we define an algebra structure on the vector space $L(\Sigma)$
generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an
oriented surface. This algebra has a filtration and the associated graded
algebra $L_{Gr}(\Sigma)$ is naturally a Poisson algebra. There is a Poisson
algebra homomorphism from the algebra of chord diagrams $ch(\Sigma)$ on $\Sigma
$ to $L_{Gr}(\Sigma)$.
  We show that multiplication in $L(\Sigma)$ provides a geometric way to define
a deformation quantization of the algebra of chord diagrams, provided there is
a universal Vassiliev invariant for links in $\Sigma\times [0,1]$. The
quantization descends to a quantization of the moduli space of flat connections
on $\Sigma $ and it is universal with respect to group homomorphisms. If
$\Sigma $ is compact with free fundamental group we construct a universal
Vassiliev invariant.


title: Skein Homology

abstract:
  For each skein module we describe a homology theory which, for any three
manifold recovers the skein module at its zero level. The theory measures
skein-like relations among skein relations, mimicking Hilbert's theory of
syzygies. We work explicit examples for the homology groups corresponding to
the Kauffman bracket. It is shown, in particular, that for every manifold most
of the groups are non-trivial.


title:
RSOS models and Jantzen-Seitz representations of Hecke algebras at roots
  of unity

abstract:
  A special family of partitions occurs in two apparently unrelated contexts:
the evaluation of 1-dimensional configuration sums of certain RSOS models, and
the modular representation theory of symmetric groups or their Hecke algebras
$H_m$. We provide an explanation of this coincidence by showing how the
irreducible $H_m$-modules which remain irreducible under restriction to
$H_{m-1}$ (Jantzen-Seitz modules) can be determined from the decomposition of a
tensor product of representations of affine $\sl_n$.


title:
Combinatorics of solvable lattice models, and modular representations of
  Hecke algebras

abstract:
  We review and motivate recently-observed relationships between exactly
solvable lattice models and modular representations of Hecke algebras. Firstly,
we describe how the set of $n$-regular partitions label both of the following
classes of objects:
  1. The spectrum of unrestricted solid-on-solid lattice models based on
level-1 representations of the affine algebras $\sl_n$,
  2. The irreducible representations of type-A Hecke algebras at roots of
unity: $H_m(\sqrt[n]{1})$.
  Secondly, we show that a certain subset of the $n$-regular partitions label
both of the following classes of objects:
  1. The spectrum of restricted solid-on-solid lattice models based on cosets
of affine algebras $(sl(n)^_1 \times sl(n)^_1)/ sl(n)^_2$.
  2. Jantzen-Seitz (JS) representations of $H_m(\sqrt[n]{1})$: irreducible
representations that remain irreducible under restriction to
$H_{m-1}(\sqrt[n]{1})$.
  Using the above relationships, we characterise the JS representations of
$H_m(\sqrt[n]{1})$ and show that the generating series that count them are
branching functions of affine $\sl_n$.


title:
Irreducible decomposition for tensor prodect representations of
  Jordanian quantum algebras

abstract:
  Tensor products of irreducible representations of the Jordanian quantum
algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest
weight finite dimensional representations of U_h(sl(2)) and lowest weight
infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product
representations are reducible and that the decomposition rules to irreducible
representations are exactly the same as those of corresponding Lie algebras.


title:
Simplified boson realization of the $so_q(3)$ subalgebra of $u_q(3)$ and
  matrix elements of $so_q(3)$ quadrupole operators

abstract:
  A simplified boson realization of the $so_q(3)$ subalgebra of $u_q(3)$ is
constructed. A simplified form of the corresponding $so_q(3)$ basis states is
obtained. The reduced matrix elements of a special second-rank tensor operator
(quadrupole operator) are calculated in the $so_q(3)$ basis.


title: The (Q,q)-Schur algebra

abstract:
  In this paper we use the Hecke algebra of type $B$ to define a new algebra
$\Sch$ which is an analogue of the q-Schur algebra. We construct Weyl modules
for $\Sch$ and obtain, as factor modules, a family of irreducible
$\Sch$-modules over any field.


title:
q-affine-Yangian double correspondence and free boson representation of
  Yangian double with arbitrary level

abstract:
  We derive a free boson representation of the Yangian double $DY_\hbar(sl_N)$
with arbitrary level k using the observation that there is a correspondence
between the q-affine algebra and Yangian double associated with the same Cartan
matrix. Vertex operator and screening currents are not obtained in the same
way.


title: Differential Calculus on $q$-Deformed Light-Cone

abstract:
  We propose the ``short'' version of q-deformed differential calculus on the
light-cone using twistor representation. The commutation relations between
coordinates and momenta are obtained. The quasi-classical limit introduced
gives an exact shape of the off-shell shifting.


title: Harmonic Oscillator Lie Bialgebras and their Quantization

abstract:
  All possible Lie bialgebra structures on the harmonic oscillator algebra are
explicitly derived and it is shown that all of them are of the coboundary type.
A non-standard quantum oscillator is introduced as a quantization of a
triangular Lie bialgebra, and a universal $R$-matrix linked to this new quantum
algebra is presented.


title: Null-plane Quantum Poincar\'e Algebras and their Universal $R$-matrices

abstract:
  A non-standard quantum deformation of the Poincar\'e algebra is presented in
a null-plane framework for 1+1, 2+1 and 3+1 dimensions. Their corresponding
universal $R$-matrices are obtained in a factorized form by choosing suitable
bases related to the $T$-matrix formalism.