Some modules over Lie algebras related to the Virasoro algebra

Abstract

In this paper, we study restricted modules over a class of $(1/2) \mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ related to the Virasoro algebra. We in fact give the classification of certain irreducible restricted $\mathfrak{g}$-modules in the sense of determining each irreducible restricted module up to an irreducible module over a subalgebra of $\mathfrak{g}$ which contains its positive part. Several characterizations of these irreducible $\mathfrak{g}$-modules are given. By the correspondence between restricted modules over $\mathfrak{g}$ and modules over the vertex algebra associated to $\mathfrak{g}$, we get the classification of certain irreducible modules over vertex algebras associated to these $\mathfrak{g}$.