Tensor product of modules over a vertex algebra

Abstract

We found a necessary and sufficient condition for the existence of the tensorproduct of modules over a vertex algebra. We defined the notion of vertexbilinear 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 thetensor product and the vertex homomorphisms. We prove the commutativity of thetensor product. We also prove the associativity of the tensor product ofmodules under certain necessary and sufficient condition. Finally, we showcertain functorial properties of the vertex homomorphims and the tensorproduct.