Representations of the q-Klein-bottle Lie algebra.

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. [ABSTRACT FROM AUTHOR]