Classification of irreducible quasifinite modules over map Virasoro algebras

TL;DR

This paper classifies all irreducible quasifinite modules over Vir  algebras, showing they are tensor products of evaluation modules and providing conditions for Verma module reducibility.

Contribution

It provides a complete classification of irreducible quasifinite modules over Vir  algebras and characterizes Verma module reducibility conditions.

Findings

All irreducible quasifinite modules are tensor products of evaluation modules.

Explicit sufficient condition for Verma module reducibility.

Necessary and sufficient conditions when A is an infinite-dimensional integral domain.

Abstract

We give a complete classification of the irreducible quasifinite modules for algebras of the form Vir \otimes A, where Vir is the Virasoro algebra and A is a Noetherian commutative associative unital algebra over the complex numbers. It is shown that all such modules are tensor products of generalized evaluation modules. We also give an explicit sufficient condition for a Verma module of Vir \otimes A to be reducible. In the case that A is an infinite-dimensional integral domain, this condition is also necessary.