The quantum dimensions of modules for vertex operator algebras are defined
and their properties are discussed. The possible values of the quantum
dimensions are obtained for rational vertex operator algebras. A criterion for
simple currents of a rational vertex operator algebra is given. A full Galois
theory for rational vertex operator algebras is established using the quantum
dimensions.

The concept of polynomials in the sense of algebraic analysis, for a single
right invertible linear operator, was introduced and studied originally by D.
Przeworska-Rolewicz \cite{DPR}. One of the elegant results corresponding with
that notion is a purely algebraic version of the Taylor formula, being a
generalization of its usual counterpart, well known for functions of one
variable. In quantum calculus there are some specific discrete derivations
analyzed, which are right invertible linear operators \cite{kac}.

An overview of the basic results on Macdonald(-Koornwinder) polynomials and
double affine Hecke algebras is given. We develop the theory in such a way that
it naturally encompasses all known cases. Among the basic properties of the
Macdonald polynomials we treat are the quadratic norm formulas, duality and the
evaluation formulas. This text is a provisional version of a chapter on
Macdonald polynomials for volume 5 of the Askey-Bateman project, entitled
"Multivariable special functions".

In this paper a general van Est type isomorphism is established. The
isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie
algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a
one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules
over the total Lie algebra and SAYD modules over the associated Hopf algebra.
In contrast to the non-general case done in our previous work, here the van Est
isomorphism is found at the first level of a natural spectral sequence, rather
than at the level of complexes.

The goal of this work is to describe a categorical formalism for (Extended)
Topological Quantum Field Theories (TQFTs) and present them as functors from a
suitable category of cobordisms with corners to a linear category, generalizing
2d open-closed TQFTs to higher dimensions. The approach is based on the notion
of an n-fold category by C. Ehresmann, weakened in the spirit of monoidal
categories (associators, interchangers, Mac Lane's pentagons and hexagons), in
contrast with the simplicial (weak Kan and complete Segal) approach of Jacob
Lurie.

We determine fusion rules (dimensions of the space of intertwining operators)
among simple modules for the vertex operator algebra obtained as an even part
of the symplectic fermionic vertex operator superalgebra. By using these fusion
rules we show that the fusion algebra of this vertex operator algebra is
isomorphic to the group algebra of the Klein four group over Z.

We construct new monomial quasi-particle bases of Feigin-Stoyanovsky's type
subspaces for affine Lie algebra $\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}$
from which the known fermionic-type formulas for $(k,3)$-admissible
configurations follow naturally. In the proof we use vertex operator algebra
relations for standard modules and coefficients of intertwining operators.

This paper provides a general operadic definition for the notion of splitting
the operations of algebraic structures. This construction is proved to be
equivalent to some Manin products of operads and it is shown to be closely
related to Rota-Baxter operators. Hence, it gives a new effective way to
compute Manin black products. The present construction is shown to have
symmetry properties.

In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak
sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite
rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak
sl_{\infty})$ admits both a Hopf algebra structure and a triangular
decomposition. In particular, it can be realized as the Drinfeld double of it's
certain Hopf subalgebras.

We define holomorphic structures on canonical line bundles of the quantum
projective space $\qp^{\ell}_q$ and identify their space of holomorphic
sections. This determines the quantum homogeneous coordinate ring of the
quantum projective space. We show that the fundamental class of $\qp^{\ell}_q$
is naturally presented by a twisted positive Hochschild cocycle. Finally, we
verify the main statements of Riemann-Roch formula and Serre duality for
$\qp^{1}_q$ and $\qp^{2}_q$.

We consider a parameter-dependent version of the homotopy associative part of
the Lian-Zuckerman homotopy algebra and provide the interpretation of
multilinear operations of this algebra in terms of integrals over certain
polytopes. We explicitly prove the pentagon relation up to homotopy and propose
a construction of higher operations.

It has previously been shown that, at least for non-exceptional Kac-Moody Lie
algebras, there is a close connection between Demazure crystals and tensor
products of Kirillov-Reshetikhin crystals. In particular, certain Demazure
crystals are isomorphic as classical crystals to tensor products of
Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we
show that this isomorphism intertwines the natural affine grading on Demazure
crystals with a combinatorially defined energy function.

We introduce a new family of twisted generalized Weyl algebras, called
multiparameter twisted Weyl algebras, for which we parametrize all simple
quotients of a certain kind. Both Jordan's simple localization of the
multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl
algebra are special cases of this construction. We classify all simple weight
modules over any multiparameter twisted Weyl algebra.

We show that the space of logarithmic intertwining operators among
logarithmic modules for a vertex operator algebra is isomorphic to the space of
3-point conformal blocks over the projective line. This is considered as a
generalization of Zhu's result for ordinary intertwining operators among
ordinary modules.

Let G be a simple algebraic group. Labelled trivalent graphs called webs can
be used to product invariants in tensor products of minuscule representations.
For each web, we construct a configuration space of points in the affine
Grassmannian. Via the geometric Satake correspondence, we relate these
configuration spaces to the invariant vectors coming from webs. In the case G =
SL(3), non-elliptic webs yield a basis for the invariant spaces.

We extend the notion of an ambidextrous trace on an ideal (developed by the
first two authors) to the setting of a pivotal category. We show that under
some conditions, these traces lead to invariants of colored spherical graphs
(and so to modified 6j-symbols).

Let $V$ be a strongly regular vertex operator algebra. For a state $h \in
V_1$ satisfying appropriate integrality conditions, we prove that the space
spanned by the trace functions Tr$_Mq^{L(0)-c/24}\zeta^{h(0)} ($M$ a
$V$-module) is a vector-valued weak Jacobi form of weight 0 and a certain index
$<h, h >/2$. We discuss refinements and applications of this result when $V$ is
holomorphic, in particular we prove that if $g = e^{h(0)}$ is a finite order
automorphism then Tr$_V q^{L(0)-c/24}g$ is a modular function of weight 0 on a
congruence subgroup of $SL_2(Z)$.

In this paper, we prove Khovanov-Lauda's cyclotomic categorification
conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the
quantum group associated with a symmetrizable Cartan datum and let $V(\Lambda)$
be the irreducible highest weight $U_q(g)$-module with a dominant integral
highest weight $\Lambda$. We prove that the cyclotomic Khovanov-Lauda-Rouquier
algebra $R^{\Lambda}$ gives a categorification of $V(\Lambda)$.

We prove that any fusion category over $\mathbb{C}$ with exactly one
non-invertible simple object is spherical. Furthermore, we classify all such
categories that come equipped with a braiding.

We give an abstract construction, based on the Belavin-Polyakov-Zamolodchikov
equations, of a family of vertex operator algebras of rank $26$ associated to
the modified regular representations of the Virasoro algebra. The vertex
operators are obtained from the tensor products of intertwining operators for a
pair of Virasoro algebras. We explicitly determine the structure coefficients
that yield the axioms of VOAs. In the process of our construction, we obtain
new hypergeometric identities.

We study Hochschild (co)homology groups of the Dunkl operator quantization of
$\Z_2$-singularity constructed by Halbout and Tang. Further, we study traces on
this algebra and prove a local algebraic index formula.

In this article, we study the moonshine vertex operator algebra starting with
the tensor product of three copies of the vertex operator algebra
$V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$
associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups,
we show the transitivity of the automorphism group of the moonshine vertex
operator algebra on the set of all full vertex operator subalgebras isomorphic
to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the
stabilizer of such a vertex operator subalgebra.

We give a characterization of Drinfeld centers of fusion categories as
non-degenerate braided fusion categories containing a Lagrangian algebra.
Further we study the quotient of the monoid of non-degenerate braided fusion
categories modulo the submonoid of the Drinfeld centers and show that its
formal properties are similar to those of the classical Witt group.

Explicit generating sets are found for all primitive ideals in the generic
quantized coordinate rings of the 3x3 special and general linear groups over an
arbitrary algebraically closed field. (Previously, generators were only known
up to certain localizations.) The generating sets form polynormal regular
sequences, from which it follows that all primitive factor algebras of these
quantized coordinate rings are Auslander-Gorenstein and Cohen-Macaulay.

From N-tensor powers of the Toeplitz algebra, we construct a multipullback
C*-algebra that is a noncommutative deformation of the complex projective space
CP(N). Using Birkhoff's Representation Theorem, we prove that the lattice of
kernels of the canonical projections on components of the multipullback
C*-algebra is free. This shows that our deformation preserves the freeness of
the lattice of subsets generated by the affine covering of the complex
projective space.

We propose a ribbon braided category approach to zeta-functions in
$q$-deformed geometry. As a proof of concept we compute $\zeta_t(C^n)$ where
$C^n$ is viewed as the standard representation in the category of modules of
$U_q(sl_n)$. We conjecture that the same $\zeta_t(C^n)$ is obtained for the
$n$-dimensional representation in the category of $U_q(sl_2)$ modules. We show
that this implies the generating function for the decomposition into
irreducibles of the symmetric tensor products $S^j(V)$ for $V$ an irreducible
representation of $sl_2$.

We describe a collection of differential graded rings that categorify weight
spaces of the positive half of the quantized universal enveloping algebra of
the Lie superalgebra gl(1|2).

We introduce a cohomology theory of grading-restricted vertex algebras. To
construct the "correct" cohomologies, we consider linear maps from tensor
powers of a grading-restricted vertex algebra to "rational functions valued in
the algebraic completion of a module for the algebra," instead of linear maps
from tensor powers of the algebra to a module for the algebra.

We introduce the notions of normal tensor functor and exact sequence of
tensor categories. We show that exact sequences of tensor categories generalize
strictly exact sequences of Hopf algebras as defined by Schneider, and in
particular, exact sequences of (finite) groups. We classify exact sequences of
tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal
faithful Hopf monads on C'' and also, in terms of self-trivializing commutative
algebras in the center of C.

In this article, we give a sufficient and necessary condition for the
$C_2$-cofiniteness of the $2$-cycle permutation orbifold model $(V\otimes
V)^\sigma$ for a $C_2$-cofinite vertex operator algebra and the $2$-cycle
permutation $\sigma$ of $V\otimes V$. As an application, we show that the
$2$-cycle permutation orbifold model of the simple Virasoro vertex operator
algebra $L(c,0)$ of minimal central charge $c$ is $C_2$-cofinite.

Given a framed vertex operator algebra and a fixed Virasoro frame, one can
define a pair of binary codes, called the 1/16-code and 1/2-code. One of the
most famous examples of framed vertex operator algebras is the moonshine vertex
operator algebra V^{\natural} constructed by Frenkel-Lepowsky-Meurman, whose
full automorphism group is the Monster simple group. In this paper, we study
the 1/16-codes for the moonshine vertex operator algebra V^\natural.

In a previous paper, we showed how one can obtain from the action of a
locally compact quantum group on a type I-factor a possibly new locally compact
quantum group. In another paper, we applied this construction method to the
action of quantum SU(2) on the standard Podles sphere to obtain Woronowicz'
quantum E(2). In this paper, we will apply this technique to the action of
quantum SU(2) on the quantum projective plane (whose associated von Neumann
algebra is indeed a type I-factor).

We give a new presentation of the Drinfeld double of the elliptic Hall
algebra introduced in a previous work with I. Burban. This presentation is
similar in spirit to Drinfeld's `new realization' of quantum affine algebras.
This answers, in the case of elliptic curves, a question of Kapranov concerning
functional relations satisfied by (principal, unramified) Eisenstein series for
the groups GL(n) over a function field. It also provides proofs of some recent
conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin.

We construct a class of new Lie algebras by generalizing the one-variable Lie
algebras generated by the quadratic conformal algebras (or corresponding
Hamiltonian operators) associated to Poisson algebras and a quasi-derivation
found by Xu. These algebras can be viewed as certain twists of Xu's generalized
Hamiltonian Lie algebras. The simplicity of these algebras is completely
determined. Moreover, we construct a family of multiplicity-free
representations of these Lie algebras and prove their irreducibility.

We give a presentation in terms of generators and relations of Hopf algebras
generated by skew-primitive elements and abelian group of group-like elements
with action given via characters. This class of pointed Hopf algebras has shown
great importance in the classification theory and can be seen as generalized
quantum groups. As a consequence we get an analog presentation of Nichols
algebras of diagonal type.

To every irreducible finite crystallographic reflection group (i.e., an
irreducible finite reflection group G acting faithfully on an abelian variety
X), we attach a family of classical and quantum integrable systems on X (with
meromorphic coefficients). These families are parametrized by G-invariant
functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and
s in G is a reflection acting trivially on T. If G is a real reflection group,
these families reduce to the known generalizations of elliptic Calogero-Moser
systems, but in the non-real case they appear to be new.

We introduce a generalization of elliptic 6j-symbols, which can be
interpreted as matrix elements for intertwiners between corepresentations of
Felder's elliptic quantum group. For special parameter values, they can be
expressed in terms of multivariable elliptic hypergeometric series related to
the root system A_n. As a consequence, we obtain new biorthogonality relations
for such series.

This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz
deformation quantization for compact quantizable Kaehler manifolds. The basic
objects, concepts, and results are given. This concerns the correct
semi-classical limit behaviour of the operator quantization, the unique
Berezin-Toeplitz deformation quantization (star product), covariant and
contravariant Berezin symbols, and Berezin transform. Other related objects and
constructions are also discussed.

It is known that finite crossed modules provide premodular tensor categories.
These categories are in fact modularizable. We construct the modularization and
show that it is equivalent to the module category of a finite Drinfeld double.

The T-systems and Y-systems are classes of algebraic relations originally
associated with quantum affine algebras and Yangians.

We give explicit formulae of Whittaker vectors for Virasoro algebra in terms
of the Jack symmetric polynomials. Our fundamental tools are the Feigin-Fuchs
bosonization and the split expression of the Calogero-Sutherland model given by
Awata-Matsuo-Odake-Shiraishi.

A braided bialgebra is called primitively generated if it is generated as an
algebra by its space of primitive elements. We prove that any primitively
generated braided bialgebra is isomorphic to the universal enveloping algebra
of its infinitesimal braided Lie algebra, notions hereby introduced. This
result can be regarded as a Milnor-Moore type theorem for primitively generated
braided bialgebras and leads to the introduction of a concept of braided Lie
algebra for an arbitrary braided vector space.

For a classical simple algebraic group $G$ we obtain the affirmative answer
for the conjecture in [8] that there exists an isomorphism between the
geometric crystal on the flag variety and the one on the unipotent subgroup
$U^-$.

Let $V$ be a simple VOA of CFT-type satisfying $V'\cong V$ and $\sigma$ a
finite automorphism of $V$. We prove that if all $V$-modules are completely
reducible and a fixed point subVOA $V^\sigma$ is $C_2$-cofinite, then all
$V^\sigma$-modules are completely reducible and every simple
$V^{\sigma}$-module appears in some twisted or ordinary $V$-modules as a
$V^{\sigma}$-submodule. We also prove that $V_L^{\sigma}$ is $C_2$-cofinite for
any lattice VOA $V_L$ and $\sigma\in \Aut(V_L)$ lifted from any triality
automorphism of $L$.

For a braided vector space $(V,\sigma)$ with braiding $\sigma$ of Hecke type,
we introduce three associative algebra structures on the space
$\oplus_{p=0}^{M}\mathrm{End}S_\sigma^p(V)$ of graded endomorphisms of the
quantum symmetric algebra $S_\sigma(V)$. We use the second product to construct
a new trace. This trace is an algebra morphism with respect to the third
product. In particular, when $V$ is the fundamental representation of
$\mathcal{U}_{q}\mathfrak{sl}_{N+1}$ and $\sigma$ is the action of the
$R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$.

In this paper we revisit and extend the work done by Chaturvedu et al, as
well as Dabrowski and Parashar. The basic premise is to take a deformed
coordinate system and give is a concrete realization. This realization is given
by a parameter of q = exp (it). Expanding in powers of 't' and applying a
deformed quantum Hamiltonian to a Free Particle yields a magnetic field. To
first order we recover a constant magnetic field. To second order we recover an
anisotropic magnetic field with an additional term.

Some filtrations of the tensor product of a highest weight module and a
lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in
\cite{LZ:2009} and one can use them to define some ideals of the modified
quantized enveloping algebra. It is shown that the quotient algebras inherit
canonical bases from the modified quantized enveloping algebra and are dual to
the quantum coordinate ring defined by Kashiwara for symmetrizable Kac-Moody
algebra $\mathfrak g$.

We describe the Szeg\"o kernel on a higher genus Riemann surface in terms of
Szeg\"o kernel data coming from lower genus surfaces via two explicit sewing
procedures where either two Riemann surfaces are sewn together or a handle is
sewn to a Riemann surface. We consider in detail the examples of the Szeg\"o
kernel on a genus two Riemann surface formed by either sewing together two
punctured tori or by sewing a twice-punctured torus to itself. We also consider
the modular properties of the Szeg\"o kernel in these cases.

In this paper we construct bases of standard modules L(Lambda) for affine Lie
algebra of type B_2^(1) consisting of semi-infinite monomials. The main
technical ingredient is a construction of monomial bases for
Feigin-Stoyanovsky's subspaces W(Lambda) of L(Lambda) by using simple currents
and intertwining operators in vertex operator algebra theory. By coincidence
W(k Lambda_0) for B_2^(1) and the standard module L(k Lambda_0) for A_1^(1)
have the same presentation P/I, so our main theorem provides a new proof of
linear independence of monomial bases of A_1^(1)-modules L(k Lambda_0).

We begin a study of the representation theory of quantum continuous
$\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends
on two parameters and is a deformed version of the enveloping algebra of the
Lie algebra of difference operators acting on the space of Laurent polynomials
in one variable. Fundamental representations of $\mathcal E$ are labeled by a
continuous parameter $u\in {\mathbb C}$.

We prove that we have an isomorphism of type $A_{aut}(\mathbb
C_\sigma[G])\simeq A_{aut}(\mathbb C[G])^\sigma$, for any finite group $G$, and
any 2-cocycle $\sigma$ on $G$. In the particular case $G=\mathbb Z_n^2$, this
leads to a Haar-measure preserving identification between the subalgebra of
$A_o(n)$ generated by the variables $u_{ij}^2$, and the subalgebra of
$A_s(n^2)$ generated by the variables $X_{ij}=\sum_{a,b=1}^np_{ia,jb}$.

A relationship between Painleve systems and infinite-dimensional integrable
hierarchies is studied. We derive a class of higher order Painleve systems from
Drinfeld-Sokolov (DS) hierarchies of type A by similarity reductions. This
result allows us to understand some properties of Painleve systems, Hamiltonian
representations, affine Weyl group symmetries and Lax forms.

The aim of this short note is to present a proof of Conjecture 1.3 of
\cite{CFFR} about the existence of an $A_\infty$-quasi-isomorphism between the
$A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule $K$, introduced in
\cite{CFFR}, and the Koszul complex $\mathrm K(V)$ of $\mathrm S(V^*)$, viewed
as a $\mathrm S(V^*)$-$\wedge(V)$-bimodule, for $V$ a finite-dimensional
(complex or real) vector space.

We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function
for the Macdonald polynomials, being constructed in the tensor product of the
ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the
degenerate $\mathbb{C} \mathbb{P}^1$. We show that a certain restriction of
$K_n(x,z;q,t)$ with respect to the variable $z$ is neatly described by the
tableau sum formula of Macdonald polynomials. Next, we demonstrate that the
integer level representation of the Ding-Iohara quantum algebra naturally
produces the currents of the deformed $\mathcal{W}$ algebra.

In this paper, we explain how the structures of affine E_7 and E_6 diagrams
can be encoded inside the Babymonster and the largest Fischer 3-transposition
group, respectively, via the theory of vertex operator algebras. We also
explain how in this framework McKay's E_7 and E_6 observations can be
understood. The main idea is to make use of certain inductive structures
associated to the Moonshine vertex operator algebra and its subalgebras related
to the Babymonster and the Fischer group.

On the vertex operator algebra associated with rank one lattice we derive a
general formula for products of vertex operators in terms of generalized
homogeneous symmetric functions. As an application we realize Jack symmetric
functions of rectangular shapes as well as marked rectangular shapes.

We construct all fundamental modules for the two parameter quantum affine
algebra of type $A$ using a combinatorial model of Young diagrams. In
particular we also give a fermionic realization of the two-parameter quantum
affine algebra.

We study coquasitriangular pointed Majid algebras via the quiver approaches.
The class of Hopf quivers whose path coalgebras admit coquasitriangular Majid
algebras is classified. The quiver setting for general coquasitriangular
pointed Majid algebras is also provided. Through this, some examples and
classification results are obtained.

We use combinatorial description of bases of Feigin-Stoyanovsky's type
subspaces of standard modules of level 1 for affine Lie algebras of types
$A_\ell^{(1)}$ and $D_4^{(1)}$ to obtain character formulas. These descriptions
naturally lead to systems of recurrence relations for which we also find
solutions.

Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion
category can be completely defined over a cyclotomic field. We show that this
is not the case: in particular one of the fusion categories coming from the
Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly
constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely
defined over a cyclotomic field.

Exceptional modular invariants for the Lie algebras B2 (at levels 2,3,7,12)
and G2 (at levels 3,4) can be obtained from conformal embeddings. We determine
the associated alge bras of quantum symmetries and discover or recover, as a
by-product, the graphs describing exceptional quantum subgroups of type B2 or
G2 which encode their module structure over the associated fusion category.
Global dimensions are given.

We give a fast algorithm for computing the canonical basis of an irreducible
highest-weight module for $U_q(\hat{\mathfrak{sl}}_e)$, generalising the LLT
algorithm.

We study the injective envelopes of the simple right $C$-comodules, and their
duals, where $C$ is a coalgebra. This is used to give a short proof and to
extend a result of Iovanov on the dimension of the space of integrals on
coalgebras. We show that if $C$ is right co-Frobenius, then the dimension of
the space of left $M$-integrals on $C$ is $\leq {\rm dim}M$ for any left
$C$-comodule $M$ of finite support, and the dimension of the space of right
$N$-integrals on $C$ is $\geq {\rm dim}N$ for any right $C$-comodule $N$ of
finite support.

We define the Hall algebra associated to any triangulated category under some
finiteness conditions with the $t$-periodic translation functor $T$ for odd
$t>1.$ This generalizes the results in \cite{Toen2005} and \cite{XX2006}.

We study admissibility conditions for the parameters of degenerate cyclotomic
BMW algebras. We show that the u-admissibility condition of Ariki, Mathas and
Rui is equivalent to a simple module theoretic condition.

In this paper, we compute the center of the infinitesimal Hecke algebras Hz
associated to sl_2 ; then using nontriviality of the center, we study
representations of these algebras in the framework of the BGG category O. We
also discuss central elements in infinitesimal Hecke algebras over gl(n) and
sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz.

The Macdonald polynomials with prescribed symmetry are obtained from the
nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation,
$t$-antisymmetrisation and normalisation. Motivated by corresponding results in
Jack polynomial theory we proceed to derive an expansion formula and a related
normalisation. Eigenoperator methods are used to relate the symmetric and
antisymmetric Macdonald polynomials, and we discuss how these methods can be
extended to special classes of the prescribed symmetry polynomials in terms of
their symmetric counterpart.

Cherednik's quantum affine Knizhnik-Zamolodchikov equations associated to an
affine Hecke algebra module M form a holonomic system of q-difference equations
acting on M-valued functions on a complex torus T. In this paper the quantum
affine Knizhnik-Zamolodchikov equations are related to the Cherednik-Macdonald
theory when M is induced from a character of a standard parabolic subalgebra of
the affine Hecke algebra.

We consider Poisson superalgebras with constant nondegenerate bracket
realized on the smooth Grassmann-valued functions with compact supports in
R^{2n}. The deformations with even and odd deformation parameters of these
superalgebras are presented for n>1.

We prove the periodicities of the restricted T and Y-systems associated with
the quantum affine algebra of type C_r, F_4, and G_2 at any level. We also
prove the dilogarithm identities for these Y-systems at any level. Our proof is
based on the tropical Y-systems and the categorification of the cluster algebra
associated with any skew-symmetric matrix by Plamondon.

Let $k$ be a field and suppose $p, q\in k$. We prove that the two affine
Hecke algebras $H_q$ and $H_p$ of type $A_n$ are isomorphic as $k$-algebras if
and only if $p=q^{\pm 1}$.

In this notes we give details of the proofs performed with GAP of the
theorems of our paper "Pointed Hopf Algebras over the Sporadic Groups".

In this paper we describe the right coideal subalgebras containing all
group-like elements of the two-parameter quantum groups Uq(g) and uq(g), where
g is a simple Lie algebra of type G2. As a consequence, we determine that there
are precisely 60 different right coideal subalgebras containing all group-like
elements.

We develop representation theory of the rational Cherednik algebra H
associated to a finite Coxeter group W in a vector space h. It is applied to
show that, for integral values of parameter `c', the algebra H is simple and
Morita equivalent to D(h)#W, the cross product of W with the algebra of
polynomial differential operators on h.

We provide a very short approach to several fundamental results of Hopf
algebras. Besides being short, our approach is the only one to prove the
bijectivity of the antipode without using the uniqueness of the integrals of
Hopf algebras and obtain the results on existence and uniqueness of integrals
as a byproduct in a way similar to the classical theory of the Haar measure on
compact groups.

In this paper we construct a q-analogue of the Legendre transformation, where
q is a matrix of formal variables defining the phase space braidings between
the coordinates and momenta (the extensive and intensive thermodynamic
observables). Our approach is based on an analogy between the semiclassical
wave functions in quantum mechanics and the quasithermodynamic partition
functions in statistical physics. The basic idea is to go from the
q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in
thermodynamics.

Let SU_q(2) and E_q(2) be Woronowicz' q-deformations of respectively the
compact Lie group SU(2) and the non-trivial double cover of the Lie group E(2)
of Euclidian transformations of the plane. We prove that, in some sense, their
duals are `Morita equivalent locally compact quantum groups'. In more concrete
terms, we prove that the von Neumann algebraic quantum groups of `bounded
measurable functions' on SU_q(2) and E_q(2) are unitary cocycle deformations of
each other.

The bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equation corresponding
to the affine Hecke algebra $H$ of type $A_{N-1}$ is a consistent system of
$q$-difference equations which in some sense contains two families of
Cherednik's quantum affine Knizhnik-Zamolodchikov equations for meromorphic
functions with values in principal series representations of $H$. In this paper
we extend this construction of BqKZ to the case where $H$ is the affine Hecke
algebra associated to an arbitrary irreducible reduced root system.

We introduce the notions of Hopf quasigroup and Hopf coquasigroup $H$
generalising the classical notion of an inverse property quasigroup $G$
expressed respectively as a quasigroup algebra $k G$ and an algebraic
quasigroup $k[G]$. We prove basic results as for Hopf algebras, such as
anti(co)multiplicativity of the antipode $S:H\to H$, that $S^2=\id$ if $H$ is
commutative or cocommutative, and a theory of crossed (co)products.

We give an interpretation of the t=1 specialization of the modified Macdonald
polynomial as a generating function of the energy statistics defined on the set
of paths arising in the context of Box-Ball Systems (BBS-paths for short). We
also introduce one parameter generalization of the energy statistics on the set
of BBS-paths which all, conjecturally, have the same distribution.

Many quantum groups and quantum spaces of interest can be obtained by cochain
(but not cocycle) twist from their corresponding classical object. This failure
of the cocycle condition implies a hidden nonassociativity in the
noncommutative geometry already known to be visible at the level of
differential forms. We extend the cochain twist framework to connections and
Riemannian structures and provide examples including twist of the $S^7$
coordinate algebra to a nonassociative hyperbolic geometry in the same category
as that of the octonions.

In this paper we use the viewpoint of the formal calculus underlying vertex
operator algebra theory to study certain aspects of the classical umbral
calculus and we introduce and study certain operators generalizing the
classical umbral shifts. We begin by calculating the exponential generating
function of the higher derivatives of a composite function, following a short,
elementary proof which naturally arose as a motivating computation related to a
certain crucial "associativity" property of an important class of vertex
operator algebras.

In 2000, Jones found two copies of the cyclic category in the annular
Temperley-Lieb category ATL. We give an abstract presentation of ATL to discuss
how these two copies of the cyclic category generate ATL together with the
coupling constants and the coupling relations. We then discuss modules over the
annular category and homologies of such modules, the latter of which arises
from the cyclic viewpoint.

We give a quick method of constructing strong homotopy associative algebras.
This method is an associative version of (higher) derived bracket construction
in the category of Lie/Leibniz algebras. We try to unify the two derived
bracket constructions. For that aim we introduce a new type of algebra ``Loday
pair", which is a noncommutative version of classical Leibniz pair. We give a
coalgebra description of Loday pairs and study a derived bracket construction
for Loday pairs.

In this paper, we give a polytopal estimate of Mirkovi\'c-Vilonen polytopes
lying in a Demazure crystal in terms of Minkowski sums of extremal
Mirkovi\'c-Vilonen polytopes. As an immediate consequence of this result, we
provide a necessary (but not sufficient) polytopal condition for a
Mirkovi\'c-Vilonen polytope to lie in a Demazure crystal.

We prove a general statement which implies the coincidence of the Azumaya and
smooth loci of the center of an algebra in positive characteristic, provided
that the spectrum of its associated graded algebra has a large symplectic leaf.
In particular, we show that for a symplectic reflection algebra smooth and the
Azumaya loci coincide.

Quantum principal bundles or principal comodule algebras are re-interpreted
as principal bundles within a framework of Synthetic Noncommutative
Differential Geometry. More specifically, the notion of a noncommutative
principal bundle within a braided monoidal category is introduced and it is
shown that a noncommutative principal bundle in the category opposite to the
category of vector spaces is the same as a faithfully flat Hopf-Galois
extension.

We define the partition and $n$-point functions for a vertex operator algebra
on a genus two Riemann surface formed by sewing two tori together. We obtain
closed formulas for the genus two partition function for the Heisenberg free
bosonic string and for any pair of simple Heisenberg modules. We prove that the
partition function is holomorphic in the sewing parameters on a given suitable
domain and describe its modular properties for the Heisenberg and lattice
vertex operator algebras and a continuous orbifolding of the rank two fermion
vertex operator super algebra.

We study Hom-quantum groups, their representations, and module Hom-algebras.
Two Twisting Principles for Hom-type algebras are formulated, and construction
results are proved following these Twisting Principles. Examples include
Hom-quantum n-spaces, Hom-quantum enveloping algebras of Kac-Moody algebras,
Hom-Verma modules, and Hom-type analogs of U_q(sl_2)-module-algebra structures
on the quantum planes.

We study locally finite simple Lie algebras containing a maximal toral
subalgebra and give the structure of those such algebras which are of countable
dimension with finite dimensional weight spaces.

We define a notion of tensor product of bimodule categories and prove that
with this product the 2-category of C-bimodule categories for fixed tensor C is
a monoidal 2-category in the sense of Kapranov and Voevodsky. We then provide a
monoidal-structure preserving 2-equivalence between the 2-category of
C-bimodule categories and Z(C)-module categories (module categories over the
center). For finite group G we show that de-equivariantization is equivalent to
tensor product over category Rep(G) of finite dimensional representations.

We give a representation-theoretic interpretation of the Langlands character
duality of Frenkel and Hernandez, and show that the "Langlands branching
multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain
tensor product multiplicities. For finite type quantum groups, the connection
with tensor products can be explained in terms of tilting modules.

We prove that the operad of framed little 2-discs is formal. Tamarkin and
Kontsevich each proved that the unframed 2-discs operad is formal. The unframed
2-discs is an operad in the category of S^1-spaces, and the framed 2-discs
operad can be constructed from the unframed 2-discs by forming the operadic
semidirect product with the circle group.

We give several necessary and sufficient conditions for the existence of {\it
the presentation by conjugation} for a non-simply laced extended affine Weyl
group. We invent a computational tool by which one can determine simply the
existence of the presentation by conjugation for an extended affine Weyl group.
As an application, we determine the existence of the presentation by
conjugation for a large class of extended affine Weyl groups.

In previous work, to each Hopf algebra H and each invertible right
two-cocycle on H, Eli Aljadeff and the first-named author attached a subalgebra
B of the free commutative Hopf algebra S generated by the coalgebra underlying
H; the algebra B is the subalgebra of coinvariants of a generic Hopf Galois
extension. In this paper we give conditions under which S is faithfully flat,
or even free, as a B-module. We also show that B is generated as an algebra by
certain elements arising from the theory of polynomial identities for comodule
algebras developped jointly with Aljadeff.

We introduce an epsilon system on a geometric crystal of type $A_n$, which is
a certain set of rational functions with some conditions. We shall show that
there is a product structure and that it is invariant under the action of
tropical R maps.

We prove an integral version of the Schur--Weyl duality between the
specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum
algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we
deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground
rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free
modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006),
138--174] in the symplectic case.

Using equivariant localization formulas we give a formula for conformal
blocks at level one on the sphere as suitable polynomials. Using this
presentation we give a generating set in the space of conformal blocks at any
level if the marked points on the sphere are generic.

For each N > 3, we define a monoidal functor from Elias and Khovanov's
diagrammatic version of Soergel's category of bimodules to the category of
sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these
functors Soergel's category can be obtained from the sl(N) foams.

We define two functors from Elias and Khovanov's diagrammatic Soergel
category, one targeting Clark-Morrison-Walker's category of disoriented sl(2)
cobordisms and the other the category of (universal) sl(3) foams.

We introduce a notion of strongly C^{\times}-graded, or equivalently,
C/Z-graded generalized g-twisted V-module associated to an automorphism g, not
necessarily of finite order, of a vertex operator algebra. We also introduce a
notion of strongly C-graded generalized g-twisted V-module if V admits an
additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a
vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let
u be an element of V of weight 1 such that L(1)u=0.

Let $K$ be a differential field over $\C$ with derivation $D$, $G$ a finite
linear automorphism group over $K$ which preserves $D$, and $K^G$ the fixed
point subfield of $K$ under the action of $G$. We show that every
finite-dimensional vertex algebra $K^G$-module is contained in some twisted
vertex algebra $K$-module.

We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms
of actions of elements in the rational and trigonometric Cherednik algebras of
type GL2 on certain subspaces of their polynomial representations. This is used
to compute the nilpotency index of the Jordanian r-matrices, thus answering a
question of Gerstenhaber and Giaquinto. We also give an interpretation of the
Cremmer-Gervais quantization in terms of the corresponding double affine Hecke
algebra.

In this paper we introduce a trace-like invariant for the irreducible
representations of a finite dimensional complex Hopf algebra H. We do so by
considering the trace of the map induced by the antipode S on the endomorphisms
End(V) of a self-dual module V. We also compute the values of this trace for
the representations of two non-semisimple Hopf algebras: u_q(sl_2) and
D(H_n(q)), the Drinfeld double of the Taft algebra.

We show that a projective space P^\infty(Z/2) endowed with the Alexandrov
topology is a classifying space for finite closed coverings of compact quantum
spaces in the sense that any such a covering is functorially equivalent to a
sheaf over this projective space. In technical terms, we prove that the
category of finitely supported flabby sheaves of algebras is equivalent to the
category of algebras with a finite set of ideals that intersect to zero and
generate a distributive lattice.

Given a family $\F$ of posets closed under disjoint unions and the operation
of taking convex subposets, we construct a category $\C_{\F}$ called the
\emph{incidence category of $\F$}. This category is "nearly abelian" in the
sense that all morphisms have kernels/cokernels, and possesses a symmetric
monoidal structure akin to direct sum. The Ringel-Hall algebra of $\C_{\F}$ is
isomorphic to the incidence Hopf algebra of the collection $\P(\F)$ of order
ideals of posets in $\F$. This construction generalizes the categories
introduced by K.

Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules
and its decomposition into irreducible modules. The gl_{N+1} Gaudin model
assigns to each multiplicity space of that decomposition a commutative (Bethe)
algebra of linear operators acting on the multiplicity space. The Bethe ansatz
method is a method to find eigenvectors and eigenvalues of the Bethe algebra.
One starts with a critical point of a suitable (master) function and constructs
an eigenvector of the Bethe algebra.

Let Uq(ghat) be the quantum affine algebra associated to a simply-laced
simple Lie algebra g. We examine the relationship between Dorey's rule, which
is a geometrical statement about Coxeter orbits of g-weights, and the structure
of q-characters of fundamental representations V_{i,a} of Uq(ghat). In
particular, we prove, without recourse to the ADE classification, that the rule
provides a necessary and sufficient condition for the monomial 1 to appear in
the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.

It is known that the notion of graded differential algebra coincides with the
notion of monoid in the monoidal category of complexes. By using the monoidal
structure introduced by M. Kapranov for the category of $N$-complexes we define
the corresponding generalization of graded differential algebras as the monoids
of this category. It turns out that this generalization coincides with the
notion of graded $q$-differential algebra which has been previously introduced
and studied.

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra
(quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes",
i.e. the elements of zero conformal weight, of the semi-infinite (BRST)
cohomology complex of the Virasoro algebra associated with vertex operator
algebra (VOA) with a formal parameter. We also construct a certain deformation
of the BRST differential parametrized by a constant two-component tensor, such
that it leads to the deformation of the $A_{\infty}$ subalgebra of the
quasiclassical LZ algebra.

The half-liberated orthogonal group $O_n^*$ appears as intermediate quantum
group between the orthogonal group $O_n$, and its free version $O_n^+$. We
discuss here its basic algebraic properties, and we classify its irreducible
representations. The classification of representations is done by using a
certain twisting-type relation between $O_n^*$ and $U_n$, a non abelian
discrete group playing the role of weight lattice for $O_n^*$, and a number of
methods inspired from the theory of Lie algebras.

This is an exposition in order to give an explicit way to understand (1) a
non-topological proof for an existence of a base of an affine root system, (2)
a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) the
isotropic root multiplicities of those elliptic Lie algebras.

This paper is devoted to the study of the quasitriangularity of Hopf algebras
via Hopf quiver approaches. We give a combinatorial description of the Hopf
quivers whose path coalgebras give rise to coquasitriangular Hopf algebras.
With a help of the quiver setting, we study general coquasitriangular pointed
Hopf algebras and obtain a complete classification of the finite-dimensional
ones over an algebraically closed field of characteristic 0.

We show how to integrate a morphism of 2-term dglas to a weak morphism of Lie
2-groups. To do so we develop a theory of butterflies of 2-term L_infty
algebras. In particular, we obtain a new description of the bicategory of
2-term L_infty algebras.

We introduce two kinds of gauge invariants for any finite-dimensional Hopf
algebra H. When H is semisimple over C, these invariants are respectively, the
trace of the map induced by the antipode on the endomorphism ring of a
self-dual simple module, and the higher Frobenius-Schur indicators of the
regular representation. We further study the values of these higher indicators
in the context of complex semisimple quasi-Hopf algebras H.

We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a
field and suppose $p, q\in k^*$ are not roots of unity. We prove that the two
quantum groups $U_q(\mathfrak {sl}_2)$ and $U_p(\mathfrak{sl}_2)$ over a field
$k$ are isomorphic as $k$-algebras if and only if $p=q^{\pm 1}$. We also
describe the group of all $k$-automorphisms of $U_q(\mathfrak{sl}_2)$ and prove
that $\text{Aut}_k(U_q(\mathfrak {sl}_2))$ is isomorphic to
$\text{Aut}_k(U_p(\mathfrak {sl}_2))$

We use the Thom-Whitney construction to show that infinitesimal deformations
of a coherent sheaf F are controlled by the differential graded Lie algebra of
global sections of an acyclic resolution of the sheaf End(E), where E is any
locally free resolution of F. In particular, one recovers the well known fact
that the tangent space to Def_F is Ext^1(E,E), and obstructions are contained
in Ext^2(E,E).

This is a historical note. In 1981 we constructed a discrete version of
quantum nonlinear Schroedinger equation. This led to our discovery of quantum
determinant: it appeared in construction of anti-pod (11). Later these became
important in quantum groups: it describes the center of Yang-Baxter algebra.
Our paper was published in Doklady Akademii Nauk vol 259, page 76 (July l981)
in Russian language.

It is shown that the question raised in Section 5.7 of [1] has an affirmative
answer.

In contrast to the classical and semiclassical settings, the Coxeter element
(12...n) which cycles the columns of an mxn matrix does not determine an
automorphism of the quantum grassmannian. Here, we show that this cycling can
be obtained by defining a cocycle twist. A consequence is that the torus
invariant prime ideals of the quantum grassmannian are permuted by the action
of the Coxeter element (12...n); we view this as a quantum analogue of the
recent result of Knutson, Lam and Speyer that the Lusztig strata of the
classical grassmannian are permuted by (12...n).

In this paper we show that the value at zero of the zeta function of the
Laplacian on the non-commutative two torus, endowed with its canonical
conformal structure, is independent of the choice of the volume element (Weyl
factor) given by a (non-unimodular) state. We had obtained, in the late
eighties, in an unpublished computation, a general formula for this value at
zero involving modified logarithms of the modular operator of the state.

A general method for constructing logarithmic modules in vertex operator
algebra theory is presented. By utilizing this approach, we give explicit
vertex operator construction of certain indecomposable and logarithmic modules
for the triplet vertex algebra W(p) and for other subalgebras of lattice vertex
algebras and their N=1 super extensions.

This paper use Parseval's theorem on Fourier series to solve the equation
$e^\tau = q$ for $\tau$ a Laurent series in $q$. It then states, as a
conjecture, an extension of this result to knots. The extension is that the
Vassiliev-Kontsevich invariants of a knot can be lifted to convergent sums of
knots in such a way that each knot is isotopic to the sum of its
Vassiliev-Kontsevich invariants. The proof of such a result seems to require a
Plancherel theorems for braid groups

Analogue to commutants in the theory of associative algebras, one can
construct a new subalgebra of vertex algebra known as a vertex algebra
commutant. In this paper, for the adjoint representation $V$ of Lie algebra
$sl(2,\C)$, we describe a commutant of $\beta\gamma$- System $S(V)$ by giving
its generators, moreover, we get a new Howe pair of vertex algebras.

We show the intimate connection between various mathematical notions that are
currently under active investigation: a class of Garside monoids, with a "nice"
Garside element, certain monoids $S$ with quadratic relations, whose monoidal
algebra $A= k[S]$ has a Frobenius Koszul dual $A^{!}$ with regular socle, the
monoids of skew-polynomial type (or equivalently, binomial skew-polynomial
rings) which were introduced and studied by the author and in 1995 provided a
new class of Noetherian Artin-Schelter regular domains, and the square-free
set-theoretic solutions of the Yang-Baxter equation.

The T-systems and Y-systems are classes of algebraic relations originally
associated with quantum affine algebras and Yangians. Recently the T-systems
were generalized to quantum affinizations of a wide class of quantum Kac-Moody
algebras by Hernandez. In this note we introduce the corresponding Y-systems
and establish a relation between T and Y-systems. We also introduce the T and
Y-systems associated with a class of cluster algebras, which include the former
T and Y-systems of simply laced type as special cases.

The leitmotif of these Notes is the idea of a vertex operator algebra (VOA)
and the relationship between VOAs and elliptic functions and modular forms.
This is to some extent analogous to the relationship between a finite group and
its irreducible characters; the algebraic structure determines a set of
numerical invariants, and arithmetic properties of the invariants provides
feedback in the form of restrictions on the algebraic structure. One of the
main points of these Notes is to explain how this works, and to give some
reasonably interesting examples.

We show how Viennot's combinatorial theory of orthogonal polynomials may be
used to generalize some recent results of Sukumar and Hodges on the matrix
entries in powers of certain operators in a representation of su(1,1). Our
results link these calculations to finding the moments and inverse polynomial
coefficients of certain Laguerre polynomials and Meixner polynomials of the
second kind.

This paper deals with two aspects of the theory of characteristic classes of
star products: first, on an arbitrary Poisson manifold, we describe Morita
equivalent star products in terms of their Kontsevich classes; second, on
symplectic manifolds, we describe the relationship between Kontsevich's and
Fedosov's characteristic classes of star products.

We introduce the concept of a quantum background and a functor QFT. In the
case that the QFT moduli space is smooth formal, we construct a flat quantum
superconnection on a bundle over QFT which defines algebraic structures
relevant to correlation functions in quantum field theory. We go further and
identify chain level generalizations of correlation functions which should be
present in all quantum field theories.