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Kac-Moody Lie Theory and Generalizations / Théorie de Lie de Kac-Moody et ses généralisations
(Nantel Bergeron, Yun Gao and Geanina Tudose, Organizer)

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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 Kac-Moody 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 Wisconsin-Madison, Madison, Wisconsin  53706, USA
Two-parameter quantum groups

In joint work with S. Witherspoon, we show that Takeuchi's two-parameter quantum groups are Drinfel'd (quantum) doubles. We construct an R-matrix and quantum Casimir operator for them, prove a complete reducibility result for their finite-dimensional representations, and obtain a Schur-Weyl duality theorem for tensor powers of their natural representation. This talk will survey these topics, and if time allows, will describe some connections with down-up 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, I-10123  Torino, Italy
Minimal representations of affine Kac-Moody groups

In this talk we will first describe the minimal representation degrees of affine Kac-Moody groups and demonstrate a mutual determination between the minimal representation degree and the root system of an affine Kac-Moody group. A classification of affine Kac-Moody groups then will follow as a consequence of the above reciprocal property. We will also present a description for all two-dimensional representations of affine Kac-Moody 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 so-called one-point function), but it doesn't recover the very desirable Jacobi form-like character of affine Kac-Moody algebras and lattices. I will try to answer the question: how should we define the character of a VOA?

DIMITAR GRANTCHAROV, University of California-Riverside, Riverside, California  92521, USA
A classification of the irreducible weight sl(n|1)-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(n|1). In particular we introduce coherent families of sl(n|1)-modules and describe explicitly the g₀-structure of the cuspidal and admissible highest weight sl(n|1)-modules.

YANAN LIN, Xiamen

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

Verma-type modules typically contain both finite and infinite-dimensional weight spaces. Although the sub-space of finite-dimensional weight spaces of a Verma-type 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 Verma-type modules and their finite parts in appropriate categories and establish an explicit equivalence between thos categories, thus establishing a characterization of Verma-type 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 finite-dimensional 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, q-supernomials and a bijection between riggings and ribbons

q-Supernomial coefficients are finitizations of string functions of Kac-Moody 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 Kazhdan-Lusztig polynomials. Hatayama et al. give another explicit expression for the q-supernomial 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  24061-0123, USA
Combinatorics of affine crystals and nilpotent varieties

We discuss connections between crystals of type A⁽¹⁾_n-1, 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 Hall-Littlewood vertex operators. There are also connections with Macdonald polynomials and with Kazhdan-Lusztig polynomials of type A⁽¹⁾_n-1, 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 G-tori 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=Zⁿ). Such root systems are classified in the same way as in the case of extended affine root systems. Then we show that a so-called Lie G-torus has such a root system. Finally, the classification of Lie G-tori of type B_r for r > 2, up to central extensions, is described.

MIKE ZABROCKI, York University, Toronto, Ontario
A quantization of non-commutative symmetric functions

The non-commutative 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 quasi-symmetric functions are attributed to Gessel in 1983 and the non-commutative 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 Hall-Littlewood 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 Hall-Littlewood bases in the non-commutative symmetric functions that arises by abstracting a formula for the Hall-Littlewood 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=1ⁿ be an n×n matrix over F satisfying q_i,i=1, q_i,j = q_j,i^-1. The q-quantum torus F_q is the unital associative algebra over F generated by t₁^±1,...,t_n^±1 subject to the defining relations t_it_j=q_i,jt_jt_i. Define the derivations ¶_i: F_q® F_q sending t₁^k₁¼t_n^k_n to k_it₁^k₁¼t_n^k_n for i Î {1,2,...,n}. Let D be a subspace of Å_i=1ⁿF¶_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.