Representations of the q-Klein-bottle Lie algebra

Abstract

We give the necessary and sufficient conditions for all weight spaces of the highest weight irreducible module V(ϕ) of the q-Klein-bottle Lie algebra B to be finite dimensional. We prove that the irreducible highest weight module V(ϕ) equals to the Verma B-module V¯(ϕ) if and only if V(ϕ) has at least one infinite-dimensional weight space. We also give the maximal proper submodule of the Verma module V¯(ϕ) and the e-character of the irreducible highest weight B-module V(ϕ) when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra B-modules with nontrivial central charge.