Verma Modules over a Block Lie Algebra

Abstract

Let [Formula: see text] be the Lie algebra with basis {Li,j, C|i, j ∈ ℤ} and relations [Li,j, Lk,l] = ((j + 1)k - i(l + 1))Li+k, j+l + iδi, -kδj+l, -2C and [C, Li,j] = 0. It is proved that an irreducible highest weight [Formula: see text]-module is quasifinite if and only if it is a proper quotient of a Verma module. An additive subgroup Γ of 𝔽 corresponds to a Lie algebra [Formula: see text] of Block type. Given a total order ≻ on Γ and a weight Λ, a Verma [Formula: see text]-module M(Λ, ≻) is defined. The irreducibility of M(Λ, ≻) is completely determined.