Quivers with relations for symmetrizable Cartan matrices and semicanonical functions
This is a report on an ongoing project with B. Leclerc and J. Schroeer. Our aim is to extend Lusztig's constructionof a semicanonical basis for the enveloping algebra of the positive part of a symmetric Kac-Moody Lie algebra,which is in terms of the preprojective algebra of the corresponding quiver over the complex numbers, to the morenatural case of symmetrizable Kac-Moody Lie algebras. To this end we study certain quivers, which usually containloops, together with a potential and some nilpotency conditions. Most of the basic constructions carry over to thisnew setup with some modifications. In particular, the components of maximal dimension of our generalized nilpotentvarieties have the structure of a B(\infty)-crystal of the corresponding type, and we can construct semicanonical functions associated to those components. To conclude, we would have to show that the constructible functionswhich have support with positive codimension, form an ideal.In the second part we can give some more details about the proofs and discuss the case B_2, which supports ourconjecture.