Deformed Kac–Moody algebras and their representations

Abstract

A class of Lie algebras G ( A ) associated to generalized Cartan matrices A is studied. The Lie algebras G ( A ) have much simpler structure than Kac–Moody algebras, but have the same root spaces with g ( A ) . In particular, G ( A ) has an abelian subalgebra of “half size.” We show that, G ( A ) has a non-degenerate invariant symmetric bilinear form if and only if A is symmetrizable; G ( X 1 ) ≅ G ( X 2 ) if and only if the GCMs X 1 and X 2 are the same up to a permutation of rows and columns. We study the lowest (respectively highest) weight Verma module V ¯ ( λ ) (respectively V ˜ ( λ ) ) over G ( A ) , and obtain the necessary and sufficient conditions for V ¯ ( λ ) to be irreducible, and also find its maximal proper submodule when V ¯ ( λ ) is reducible. Then using graded dual module of V ¯ ( λ ) we deduce the necessary and sufficient conditions for V ˜ ( λ ) to be irreducible.