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

A well-known theorem of Mathieuʼs 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 {Lα,i,C|α,i∈Z,i⩾0} and relations [Lα,i,Lβ,j]=((i+1)β−(j+1)α)Lα+β,i+j+δα+β,0δi+j,0α3−α12C. Namely, an irreducible quasifinite BB-module is either a highest weight module, a lowest weight module or a module of the intermediate series.