Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Abstract

Let B be the Block type Lie algebra over ℂ with basis {L α, i , C 1, C 2 | (α, i) ∈ ℤ × ℤ \\ {(0, − 2)}} and Lie bracket [L α, i , L β, j ] = (β(i + 1) − α(j + 1))L α+β, i+j + αδα+β, 0δ i+j, −2 C 1 + (i + 1)δα+β, 0δ i+j, −2 C 2, where C 1, C 2 are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.