Non-weight modules over algebras related to the Virasoro algebra

TL;DR

This paper classifies certain non-weight modules over algebras related to the Virasoro algebra, including their simplicity and isomorphism classes, and explores tensor product constructions for module simplicity.

Contribution

It provides a complete classification of non-weight modules over loop-Virasoro and Block type algebras, including simplicity criteria and tensor product constructions.

Findings

Classified modules with rank-one restriction to Cartan subalgebra

Established necessary and sufficient conditions for module simplicity

Determined isomorphism classes of these modules

Abstract

In this paper, we study a class of non-weight modules over two kinds of algebras related to the Virasoro algebra, i.e., the loop-Virasoro algebras $\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where $q$ is a nonzero complex number. We determine those modules whose restriction to the Cartan subalgebra (modulo center) are free of rank one. We also provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.