Classification of quasifinite representations of a Lie algebra related to Block type

TL;DR

This paper extends Mathieu's classification of Harish-chandra modules from the Virasoro algebra to a Lie algebra related to Block type, showing that irreducible quasifinite modules are either highest, lowest, or intermediate series.

Contribution

It proves an analogous classification theorem for the Lie algebra related to Block type, identifying all irreducible quasifinite modules as highest, lowest, or intermediate series.

Findings

Irreducible quasifinite modules are classified into three types.

The classification extends Mathieu's theorem to a new Lie algebra.

The structure of modules over the algebra is fully characterized.

Abstract

A well-known theorem of Mathieu's states that a Harish-chandra module over the Virasoro algebra is either a highest weight module, a lowest weight module or a module of the intermediate series. It is proved in this paper that an analogous result also holds for the Lie algebra $\BB$ related to Block type, with basis {L_{\a,i},C|a,i\in\Z, i\ge0} and relations [L_{\a,i},L_{\b,j}]=((i+1)\b-(j+1)\a)L_{\a+\b,i+j}+\d_{\a+\b,0}\d_{i+j,0}\frac{\a^3-\a}{6}C, [C,L_{\a,i}]=0.Namely, an irreducible quasifinite $\BB$-module is either a highest weight module, a lowest weight module or a module of the intermediate series.