Tensor products of strongly graded vertex algebras and their modules

TL;DR

This paper investigates the structure of tensor products of strongly graded vertex algebras and their modules, establishing irreducibility and complete reducibility conditions that generalize known results for vertex operator algebras.

Contribution

It introduces new theorems on the irreducibility and reducibility of tensor product modules for strongly graded vertex algebras, extending existing theories.

Findings

Tensor product of strongly graded irreducible modules remains irreducible.

Certain modules for tensor product algebras are completely reducible iff factors are.

Results generalize known properties from vertex operator algebras.

Abstract

We study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules.