Representations of Kac-Moody Algebras and Combinatorics (05w5064)

Objectives

This conference is intended to bring together people who work on the combinatorial aspects of the representation theory of affine Kac--Moody algebras and quantum groups with people working on algebraic and geometric aspects of the representation theory. The interaction afforded by the meeting would stimulate contact between people in the two areas, clarify and make rigorous the connections between the two and in turn provide new insights into both subjects. In addition both subjects are closely related to important developments in other areas of mathematics and mathematical physics. For example the combinatorics of crystal bases is related to the so-called cluster algebras and the geometry of flag varieties, while the study of level zero representations and fusion products is related to solvable lattice models.

Inspired by confromal field thoery,Drinfeld, Feigin, Kazdhan, Lusztig and others defined the structure of a braided monoidal category on the category of integrable modules with fixed central change of an affine Lie algebra. What is quite remarkable about this category (usually called the fusion category) is that it is naturally equivalent to a certain category of modules over the quantized universal enveloping algebras. This relates the representation theory of affine Kac-Moody algebras to that of quantum universal enveloping algebras. One way to study this category is use the observation of Feigin that the fusion product for afffine Kac-Moody algberas induces a natural grading on irreducible components of the tensor product of irreducible modules for the underlying simple Lie algebra. This replaces the Clebsch--Gordan numbers (multiplicities of irreducible representations inside the tensor product of irreducibles) by certain polynomials. In some cases they are well known Kostka polynomials. When the number of factors in the tensor product tends to $\infty$ these polynomials become characters of corresponding affine Kac-Moody algebras. There are a number of interesting combinatorial consequences of this observation related to symmetric functions and the fermionic formulas of Kirillov and Reshetikhin.

A major development in the representation theory of quantum groups was the discover by Kashiwara and Lusztig of a natural bases for the positive part of the quantized enveloping algebra of a Kac--Moody Lie algebra. the so-called global crystal bases or canonical bases. These bases also lead to natural bases in the representations of affine Lie algebras. While, the bases associated to the positive level representations of affine Lie algebras has been extensively studied, the corresponding results for level zero representations are not well--understood as yet. It is known for instance that not all level zero representations admit such bases. Kashiwara has some conjectures which list all level zero modules which admit crystal bases, although it is not yet been proved that in fact these modules have a crystal base to begin with. Partial results due to Kashiwara are known in the case of fundamental level zero modules. The theory of such bases has a very important combinatorial ingredient known as a crystal. This has been generalized to the notion of virtual crystals by Okado, Schilling and Shimozono, the representation theoretic aspect of their objects are far from clear and should be interesting to develop.

The theory of quivers and quiver varieties was developed by H. Nakajima to study representations of Kac--Moody Lie algebras and level zero representations of quantum affine algebras. The initial geometric approach using an example of a quiver variety of type $A$, the cotangent bundle of the $n$--step partial flag variety was developed by Ginzburg and Vasserot to study these representations for quantum affine sl(n). Given any simple Lie algebra and a dominant integral weight of it, Nakajima associated to it two types of quiver varieties: one a non--singular quasi--projective variety $M$ and the other, an affine algebraic variety $N$, both having a natural group action and an equivariant projective morphism from $M$ to $N$. . Nakajima showed that a tensor product of the fundamental modules of a quantum affine algebra corresponding to the dominant integral weight could be realized as the specialized equivariant K-homology associated in a natural way with this data. There are a number of results on crystals, on decompositions of these modules as modules for the quantum algebra associated to the simple Lie algebra and on computing characters of fusion productts that can be proved using this approach for these modules. The corresponding results for the irreducible quotients of these modules are much harder to prove, although here too, there has been some progress recently, in papers of Beck, Nakajima.