Deformed Kac–Moody algebras and their representations

A class of Lie algebras associated to generalized Cartan matrices A is studied. The Lie algebras have much simpler structure than Kac–Moody algebras, but have the same root spaces with . In particular, has an abelian subalgebra of “half size.” We show that, has a non-degenerate invariant symmetric bilinear form if and only if A is symmetrizable; if and only if the GCMs X1 and X2 are the same up to a permutation of rows and columns. We study the lowest (respectively highest) weight Verma module (respectively ) over , and obtain the necessary and sufficient conditions for to be irreducible, and also find its maximal proper submodule when is reducible. Then using graded dual module of we deduce the necessary and sufficient conditions for to be irreducible.