Quasifinite representations of a family of Lie algebras of Block type

To any nonzero additive subgroup G of an algebraically closed field image of characteristic zero and s=0,1, there corresponds a Lie algebra B(s,G) of Block type, with basis {xa,iaset membership, variantG, image, and relation [xa,i,xb,j]=s(b−a)xa+b,i+j+((a−1+s)j−(b−1+s)i)xa+b,i+j−1. In this paper, it is proved that B(s,G) has a nontrivial quasifinite module if and only if s=1 and G is isomorphic to image, and that a quasifinite image-module is a highest or lowest weight module. Furthermore, the quasifinite irreducible highest weight image-modules are classified and the unitary ones are proved to be trivial.