A class of irreducible modules for loop-Virasoro algebras

TL;DR

This paper investigates the conditions under which tensor products of certain modules for loop-Virasoro algebras are irreducible and isomorphic, extending prior work on Virasoro modules.

Contribution

It provides necessary and sufficient conditions for the irreducibility and isomorphism of tensor product modules for loop-Virasoro algebras, a problem previously open.

Findings

Established criteria for irreducibility of tensor product modules.

Derived conditions for isomorphism between modules.

Extended understanding of module structure for loop-Virasoro algebra.

Abstract

Tensor product of highest weight modules and intermediate modules for Virasoro algebra have been studied around 1997. Since then the irreducibility problem for tensor product of modules is open. We consider the loop-Virasoro algebra $Vir \otimes B$, where $Vir$ is the Virasoro algebra and $B$ a commutative associative unital algebra over $\mathbb C$. In this paper we study the irreducibility problem for the tensor product of highest weight modules and intermediate modules for $Vir\otimes B$. Finally we find out a necessary and sufficient conditions for such modules to be isomorphic.