Tensor product of modules over a vertex algebra

TL;DR

This paper establishes conditions for tensor product existence over vertex algebra modules, introduces vertex bilinear maps, and proves key properties like commutativity and associativity under specific conditions.

Contribution

It provides the first necessary and sufficient criteria for tensor products of modules over vertex algebras and introduces algebraic constructions including a ring-theoretic approach.

Findings

Established necessary and sufficient conditions for tensor product existence.

Proved the commutativity of the tensor product.

Proved the associativity of the tensor product under certain conditions.

Abstract

We found a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We defined the notion of vertex bilinear map and we provide two algebraic construction of the tensor product, where one of them is of ring theoretical type. We show the relation between the tensor product and the vertex homomorphisms. We prove the commutativity of the tensor product. We also prove the associativity of the tensor product of modules under certain necessary and sufficient condition. Finally, we show certain functorial properties of the vertex homomorphims and the tensor product.