Quasifinite representations of a Lie algebra of Block type

Let be the Lie algebra of Block type over with basis {Lα,i, , i −1} and relations [Lα,i,Lβ,j]=((i+1)β−(j+1)α)Lα+β,i+j+αδα+β,0δi+j,−2C, [C,Lα,i]=0. In this paper, it is proved that a quasifinite irreducible -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.