In this talk I will discuss a recent work (arXiv:math.RT/0310314) on the relation between two realizations of crystal graphs. Crystal graphs, which can be viewed as the q=0 limit of quantum groups, reduce many questions in representation theory (such as computation of characters and decomposition of tensor products of representations into sums of indecomposable ones) to combinatorics. Crystal graphs of representations of Kac-Moody algebras can be realized geometrically on the set of irreducible components of certain varieties attached to quivers as well as on combinatorial objects such as Young tableaux and Young walls. I will discuss an explicit isomorphism between these two constuctions. Some benefits of this relationship include obtaining an explicit enumeration of irreducible components of quiver varieties by classical combinatorial objects as well as giving a geometric interpretation of the combinatorial constructions (which allows us, in some cases, to extend the combinatorial constructions to more general cases). I will review the necessary material on crystal graphs and quiver varieties.

In my talk (joint with B.Kreussler) I am going to show that the group SL(2,Z) acts on the derived category of coherent sheaves on a nodal Weierstrass cubic curve. This action establishes an equivalence between the category of semistable torsion free sheaves of degree zero and the category of finite-dimensional (k[[x,y]]/xy)-modules. It allows to get a geometric description of indecomposable torsion free semi-stable sheaves of degree zero. This approach is quite similar to the description of the derived category of a canonical tubular algebra using weighted projective lines.

This is closely related to the representation theory of preprojective algebras. This will be discussed in detail.