Tensor product weight modules over the affine-Virasoro algebra

TL;DR

This paper investigates tensor product modules over the affine-Virasoro algebra, establishing conditions for their irreducibility and isomorphism, thus introducing new classes of irreducible weight modules.

Contribution

It extends the shifting technique from the Virasoro algebra to the affine-Virasoro algebra, characterizing irreducibility and isomorphism of tensor product modules.

Findings

Derived necessary and sufficient conditions for irreducibility.

Classified when tensor product modules are isomorphic.

Constructed new irreducible weight modules.

Abstract

In this paper, we study the tensor products of irreducible highest weight modules with irreducible loop modules over the affine-Virasoro algebra with aid of the ``shifting technique" established for the Virasoro algebra in [H. Chen, X. Guo, K. Zhao, Tensor product weight modules over the Virasoro algebra, J. Lond. Math. Soc. 88(2013), 829-844.]. All such tensor product modules are indecomposable modules with infinite-dimensional weight spaces. Moreover, we obtain the necessary and sufficient conditions for such tensor product modules to be irreducible. Therefore, we obtain a class of new irreducible weight modules over the affine-Virasoro algebra. Finally, the necessary and sufficient conditions for any two such tensor product modules to be isomorphic are also determined.