




KacMoody Lie Theory and Generalizations / Théorie de Lie de
KacMoody et ses généralisations (Nantel Bergeron, Yun Gao and Geanina Tudose, Organizer)
 BRUCE ALLISON, Department of Mathematical and Statistical Sciences, University of
Alberta, Edmonton, Alberta T6G 2G1
Coordinate algebras for extended affine Lie algebras of rank 1

Extended affine Lie algebras are higher nullity generalizations of
affine KacMoody Lie algebras. The structure of extended affine Lie
algebras has been determined in many cases by means of coordinatization
theorems. In rank 1, the coordinate algebras are nonassociative. In
this talk, we will describe some recent joint work with Yoji Yoshii on
the structure of these coordinate algebras.
 GEORGIA BENKART, Department of Mathematics, University of WisconsinMadison,
Madison, Wisconsin 53706, USA
Twoparameter quantum groups

In joint work with S. Witherspoon, we show that Takeuchi's
twoparameter quantum groups are Drinfel'd (quantum) doubles. We
construct an Rmatrix and quantum Casimir operator for them, prove a
complete reducibility result for their finitedimensional
representations, and obtain a SchurWeyl duality theorem for tensor
powers of their natural representation. This talk will survey these
topics, and if time allows, will describe some connections with downup
algebras (which generalize the algebra generated by the down and up
operators on a partially ordered set).
 STEPHEN BERMAN, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6
Half lattice VOA's and toroidal algebras

Some recent work of C. Dong, S. Tan and the speaker on the
representation theory of ``half lattice'' VOA's will be presented.
This study is motivated by the representation theory of some EALA's,
especially toroidal algebras.
 YU CHEN, Department of Mathematics, University of Torino,
I10123 Torino, Italy
Minimal representations of affine KacMoody groups

In this talk we will first describe the minimal representation degrees
of affine KacMoody groups and demonstrate a mutual determination
between the minimal representation degree and the root system of an
affine KacMoody group. A classification of affine KacMoody groups
then will follow as a consequence of the above reciprocal property. We
will also present a description for all twodimensional representations
of affine KacMoody groups.
 TERRY GANNON, Alberta
The character of VOAs

As defined at present, the character of modules of vertex operator
algebras suffer a major flaw: they don't distinguish the modules. There
is an obvious solution (using the socalled onepoint function), but it
doesn't recover the very desirable Jacobi formlike character of
affine KacMoody algebras and lattices. I will try to answer the
question: how should we define the character of a VOA?
 DIMITAR GRANTCHAROV, University of CaliforniaRiverside, Riverside,
California 92521, USA
A classification of the irreducible weight sl(n1)modules

We extend Olivier Mathieu's classification of all irreducible weight
sl(n)modules with finite dimensional weight spaces to the case of
Lie superalgebra g=sl(n1). In particular we introduce coherent
families of sl(n1)modules and describe explicitly the
g_{0}structure of the cuspidal and admissible highest weight
sl(n1)modules.
 YANAN LIN, Xiamen

 DUNCAN MELVILLE, St. Lawrence University, Canton, New York 13617, USA
Categories of representations for quantum affine algebras and
certain subalgebras

Vermatype modules typically contain both finite and
infinitedimensional weight spaces. Although the subspace of
finitedimensional weight spaces of a Vermatype module is not a
representation for the whole algebra, one can construct a certain
subalgebra for which it is a module. Here, we embed both the
Vermatype modules and their finite parts in appropriate categories and
establish an explicit equivalence between thos categories, thus
establishing a characterization of Vermatype modules in terms of their
finite parts.
 ERHARD NEHER, University of Ottawa, Ottawa, Ontario
Locally finite root systems

Locally finite root systems are defined in analogy to the usual
definition of finite root systems, except that finiteness is replaced
by local finiteness: the intersection of the root system with every
finitedimensional subspace is finite. In the talk I will describe ths
structure of these root systems, with special emphasize on positive
systems and the theory of weights.
 ANNE SCHILLING, Department of Mathematics, University of California, Davis,
California 95616, USA
String functions, qsupernomials and a bijection between
riggings and ribbons

qSupernomial coefficients are finitizations of string functions of
KacMoody algebras. A combinatorial expression of these coefficients
for type A as the generating function of ribbon tableaux with
(co)spin statistics follows from work of Lascoux, Leclerc and Thibon.
Leclerc and Thibon have also established a link to certain parabolic
KazhdanLusztig polynomials. Hatayama et al. give another explicit
expression for the qsupernomial coefficients. In this talk we
interpret the formulas by Hatayama et al. in terms of rigged
configurations and provide an explicit statistic preserving bijection
between rigged configurations and ribbon tableaux thereby establishing
a new direct link between these combinatorial objects underlying the
string functions.
 MARK SHIMOZONO, Department of Mathematics, Virginia Tech, Blacksburg,
Virginia 240610123, USA
Combinatorics of affine crystals and nilpotent varieties

We discuss connections between crystals of type A^{(1)}_{n1}, spaces
of global sections of coherent sheaves on nilpotent adjoint orbit
closures in gl(n), and spaces of graded endomorphisms of the ring of
symmetric functions related to Jing's HallLittlewood vertex
operators. There are also connections with Macdonald polynomials and
with KazhdanLusztig polynomials of type A^{(1)}_{n1}, both usual
and parabolic.
 YOJI YOSHII, Department of Mathematical and Statistical Sciences, University of
Alberta, Edmonton, Alberta T6G 2G1
Root systems extended by an abelian group G, and the
classification of Lie Gtori of type B

Let G be an abelian group. We define a new notion called a root
system extended by G which generalizes an affine root system (when
G=Z) and an extended affine root system (when G=Z^{n}). Such root
systems are classified in the same way as in the case of extended
affine root systems. Then we show that a socalled Lie Gtorus has
such a root system. Finally, the classification of Lie Gtori of
type B_{r} for r > 2, up to central extensions, is described.
 MIKE ZABROCKI, York University, Toronto, Ontario
A quantization of noncommutative symmetric functions

The noncommutative and quasi symmetric functions are dual Hopf
algebras that share many of the same properties and are strongly
related to the space of symmetric functions. The quasisymmetric
functions are attributed to Gessel in 1983 and the noncommutative
symmetric functions have their origins in the early 90's, and hence are
fairly new by comparison to the space of symmetric functions. Much of
the theory that is well understood for the symmetric functions has yet
to be generalized to this pair of algebras and some interesting
questions arise in developing analogs of constructions that are well
known for the symmetric functions.
Much of the recent research in symmetric functions has been on the
properties of two remarkable bases and their generalizations, the
HallLittlewood and Macdonald symmetric functions. These bases depend
on a parameter q and by specializing the parameter to various values
they interpolate many of the well known bases of the symmetric
functions. One reason they are of interest is that algebraic
identities involving these functions often encode several well known
identities in the space of symmetric functions at the same time. We
define a possible analog to the Macdonald and HallLittlewood bases in
the noncommutative symmetric functions that arises by abstracting a
formula for the HallLittlewood functions to the level of Hopf algebras
and then demonstrate some of the surprising properties held by these
functions.
This is joint work with Nantel Bergeron.
 KAIMING ZHAO, Carleton University, Ottawa, Ontario
Quantum differential operator algebras

Quantum version of the algebra of differential operators on Laurent
polynomials are given, which can be realized also as iterated skew
polynomial rings.
Let F be a field of characteristic 0, q=(q_{i,j})_{i,j=1}^{n} be
an n×n matrix over F satisfying q_{i,i}=1, q_{i,j} = q_{j,i}^{1}. The qquantum torus F_{q} is the unital associative
algebra over F generated by t_{1}^{±1},...,t_{n}^{±1} subject to
the defining relations t_{i}t_{j}=q_{i,j}t_{j}t_{i}. Define the derivations
¶_{i}: F_{q}® F_{q} sending t_{1}^{k1}¼t_{n}^{kn} to
k_{i}t_{1}^{k1}¼t_{n}^{kn} for i Î {1,2,...,n}. Let D be
a subspace of Å_{i=1}^{n}F¶_{i}. The vector space F_{q}[D]
which consists of polynomials of elements in D with coefficients in
F_{q} and which is regarded as operators on F_{q} forms naturally an
associative algebra, this is a quatun differential operator algebra.
Properties of these algebras, as associative and Lie algebras (for
example, semisimple elements, isomorphism problem, second cohomology
group) are determined.

