Classification of Irreducible integrable modules for toroidal Lie-algebras with finite dimensional weight spaces

Abstract

Toroidal Lie algebras are universal central extentions of the finite dimensional simple Lie algbera tensored with Laurent Polynomials in several commuteing variables. In this paper we classify irreducible integrable modules for Toroidal Lie algebras with finite dimensional wieght spaces. They are tensor product of evaluation modules for the finite dimensional Lie algebra when the center acts by zero. In the case when the center acts nontrivially they are tensor product of evalution representations of the underlying affine Lie algebra twisted by an automorphism of the Toroidal Lie algebras. The automorphisms come from genaral linear group with integer integers whose determinant 1 or -1.