A class of non-weight modules over the Virasoro algebra

Abstract

For any triple $(μ,λ,α)$ of complex numbers and an $\mathfrak a$-module ${V}$, a class of non-weight modules $\mathcal{M}\big(V,μ,Ω(λ,α)\big)$ over the Virasoro algebra $\mathcal L$ is constructed in this paper. We prove if $V$ is a nontrivial simple $\mathfrak a$-module satisfying: for any $v\in V$ there exists $r\in\Z_+$ such that $L_{r+i}v=0$ for all $i\geq1$, then $\mathcal{M}\big(V,μ,Ω(λ,α)\big)$ is simple if and only if $μ\neq1, λ\neq0,α\neq0,$. We also give the necessary and sufficient conditions for two such simple $\mathcal L$-modules being isomorphic. Finally, we prove that these simple $\mathcal L$-modules $\mathcal{M}\big(V,μ,Ω(λ,α)\big)$ are new by showing they are not isomorphic to any other known simple non-weight module provided that $V$ is not a highest weight $\mathfrak a$-module with highest weight nonzero.