A class of non-weight modules over the Virasoro algebra

TL;DR

This paper constructs a new class of non-weight modules over the Virasoro algebra using specific parameters and modules, providing conditions for their simplicity and isomorphism, and demonstrating their novelty compared to existing modules.

Contribution

It introduces a new family of simple non-weight Virasoro modules parameterized by triples and modules, with criteria for simplicity and isomorphism, expanding the understanding of Virasoro representations.

Findings

Constructed a new class of non-weight modules over the Virasoro algebra.

Established necessary and sufficient conditions for module simplicity and isomorphism.

Proved these modules are new and not isomorphic to previously known modules.

Abstract

For any triple $(\mu,\lambda,\alpha)$ of complex numbers and an $\mathfrak a$-module ${V}$, a class of non-weight modules $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ 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 $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ is simple if and only if $\mu\neq1, \lambda\neq0,\alpha\neq0,$. 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 $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ 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.