Quantum Deformations of Generalized Kac-Moody Algebras and Their Modules

AbstractWe construct quantum groups Uq(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 show that, for generic q, Verma modules over U(g) with integral highest weights and irreducible highest weight modules over U(g) with dominant integral highest weights can be deformed to those over Uq(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 Uq(g) with dominant integral highest weights are given by the Weyl-Kac-Borcherds formula