Quasifinite representations of a Lie algebra of Block type

Abstract

Let B be the Lie algebra of Block type over C with basis { L α, i , C∣α,i∈ Z , i⩾−1} and relations [ L α, i , L β, j ]=(( i+1) β−( j+1) α) L α+ β, i+ j + αδ α+ β,0 δ i+ j,−2 C, [ C, L α, i ]=0. In this paper, it is proved that a quasifinite irreducible B -module is a highest or lowest weight module. Furthermore, the quasifinite irreducible highest weight modules are classified and the unitary ones are proved to be trivial.