Representations of the Local Area-Preserving Lie Algebra for the Klein Bottle

Abstract

We give the necessary and sufficient conditions for all weight subspaces of the highest weight irreducible module V(φ) of the Lie algebra ℬ related to the Klein-bottle to be finite dimensional. We prove the Verma ℬ-module is irreducible if and only if the corresponding irreducible highest weight ℬ-module V(φ) has at least one infinite dimensional weight subspace. We also give the classification of the Harish-Chandra ℬ-module with nontrivial center.