Quasifinite representations of a class of Block type Lie algebras B(q)

Intrigued by a well-known theorem of Mathieu’s on Harish-Chandra modules over the Virasoro algebra, we show that any quasifinite irreducible module over a class of Block type Lie algebras B(q) is either a highest or lowest weight module, or else a uniformly bounded module, where the parameter q is a nonzero complex number. We also classify quasifinite irreducible highest weight B(q)-modules and irreducible B(q)-modules of the intermediate series. In particular, we obtain that an irreducible B(q)-module of the intermediate series may be a nontrivial extension of a -module of the intermediate series if q is half of a negative integer, where is a subalgebra of B(q) isomorphic to the Virasoro algebra.