Simple Virasoro modules which are locally finite over a positive part

TL;DR

This paper introduces a broad construction of simple Virasoro modules that encompasses highest weight and Whittaker modules, reducing their classification to finite-dimensional solvable Lie algebra modules, and extends existing classifications to new cases.

Contribution

It presents a general framework for constructing simple Virasoro modules and extends the classification of modules over a family of solvable Lie algebras, including new modules.

Findings

Unified construction of Virasoro modules including known types

Extended classification to new solvable Lie algebra modules

Recovered and constructed new simple Virasoro modules

Abstract

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This reduces the problem of classification of simple Virasoro modules which are locally finite over a positive part to classification of simple modules over a family of finite dimensional solvable Lie algebras. For one of these algebras all simple modules are classified by R. Block and we extend this classification to the next member of the family. As a result we recover many known but also construct a lot of new simple Virasoro modules. We also propose a revision of the setup for study of Whittaker modules.