Quasifinite representations of a class of Block type Lie algebras $\BB$

TL;DR

This paper extends Mathieu's theorem to a class of Block type Lie algebras, classifies their quasifinite modules, and explores conditions under which modules extend Virasoro modules.

Contribution

It provides a classification of quasifinite irreducible modules and extends Mathieu's theorem to Block type Lie algebras with parameter q.

Findings

Classification of quasifinite irreducible highest weight modules

Identification of conditions for modules to extend Virasoro modules

Extension of Mathieu's theorem to Block type Lie algebras

Abstract

Intrigued by a well-known theorem of Mathieu's on Harish-Chandra modules over the Virasoro algebra, we give an analogous result for a class of Block type Lie algebras $\BB$, where the parameter $q$ is a nonzero complex number. We also classify quasifinite irreducible highest weight $\BB$-modules and irreducible $\BB$-modules of the intermediate series. In particular, we obtain that an irreducible $\BB$-module of the intermediate series may be a nontrivial extension of a $\Vir$-module of the intermediate series if $q$ is half of a negative integer, where $\Vir$ is a subalgebra of $\BB$ isomorphic to the Virasoro algebra.