Hom functor and the associativity of tensor products of modules for vertex operator algebras (Q1355522)

By using a geometric approach J. Lepowsky and Y.-Z. Huang developed a theory of tensor products for modules for vertex operator algebras; in [\textit{Y.-Z. Huang}, A theory of tensor products for module categories for a vertex operator algebra. IV, J. Pure Appl. Algebra 100, 173-216 (1995; Zbl 0841.17015)] the associativity of tensor products was proved under certain assumptions on rationality of vertex operator algebra and on convergence properties of products of its intertwining operators. In the paper under review the authors give, under the same assumptions, a new proof of the associativity of tensor products. Their proof does not depend on the details of the construction of the tensor product: they follow the algebraic approach where the tensor product of two \(V\)-modules \(W_1\) and \(W_2\) is defined to be a \(V\)-module \(W_{12}\) together with an intertwining operator \(F_{12}(\cdot,z)\) of the type \(\binom{W_{12}}{W_1\;W_2}\) such that for any \(V\)-module \(W\) and any intertwining operator \(I(\cdot,z)\) of the type \(\binom{W}{W_1\;W_2}\) there exists a unique \(V\)-homomorphism \(\psi\) from \(W_{12}\) to \(W\) such that \(I(\cdot,z)=\psi\circ F_{12}(\cdot,z)\) [cf. \textit{H. Li}, An analogue of ``Hom''-functor and a generalized nuclear democracy theorem, preprint].