Quantum deformations of generalized Kac-Moody algebras and their modules

Abstract

We construct quantum groups U_q(g) associated with generalized Kac-Moody algebras g with admissible Borcherds-Cartan matrices. We also construct quantum deformations of highest weight modules over U(g) with integral highest weights. We sho that, for generic q, Verma modules over U(g) with integral highest weights and irreducible highest weight modules over U(g) with dominant integral highests can be deformed to those over U_q(g) in such a way that the dimensions of weight spaces are invariant under the deformation. In particular, for generic q, the characters of irreducible highest weight modules over U_q(g) with dominant integral highest weights are given by the Weyl-Kac-Borcherds formula.