Modules induced from polynomial subalgebras of the Virasoso algebra

TL;DR

This paper introduces polynomial subalgebras of the Virasoro algebra, describes their one-dimensional modules, and studies the induced modules, revealing new simple modules and generalizing recent findings.

Contribution

It defines polynomial subalgebras of the Virasoro algebra and analyzes the structure of modules induced from these subalgebras, extending known simple modules.

Findings

Induced modules are often simple

New simple modules are constructed via tensor products

Results recover and extend recent classifications of simple Virasoro modules

Abstract

The Virasoro Lie algebra is a one-dimensional central extension of the Witt algebra, which can be realized as the Lie algebra of derivations on the algebra $\cc [t^{\pm}]$ of Laurent polynomials. Using this fact, we define a natural family of subalgebras of the Virasoro algebra, which we call polynomial subalgebras. We describe the one-dimensional modules for polynomial subalgebras, and we use this description to study the corresponding induced modules for the Virasoro algebra. We show that these induced modules are frequently simple and generalize a family of recently discovered simple modules. Additionally, we explore tensor products involving these induced modules. This allows us to describe new simple modules and also recover results on recently discovered simple modules for the Virasoro algebra.