# Abelian ideals, Koszul algebras and representations of current algebras

SPEAKER: Jacob Greenstein

TITLE: Abelian ideals, Koszul algebras and representations of current algebras

ABSTRACT In this talk (based on a joint work with V. Chari) we will discuss connections between some classical objects – abelian ideals in a Borel subalgebra and graded characters of symmetric and exterior algebras of a finite dimensional simple Lie algebra – and finite dimensional representations of polynomial current algebras which are generalizations of Kirillov-Reshetikhin modules. Namely, to each abelian ideal (satisfying some combinatorial condition), one can associate a family of finite dimensional Koszul algebras and an infinite family of indecomposable representations of polynomial current algebras which correspond to indecomposable projective modules under a natural equivalence of categories. Their graded characters can thus be expressed in terms of graded characters of the symmetric algebra of the adjoint representation. Furthermore, the inductive limit of this family of algebras is an infinite dimensional Koszul algebra and can be constructured in a “Lie-theoretically explicit” way.