The main results of a theory developed by the author and J. Lepowsky are described in [Tensor products of modules for a vertex operator algebra and vertex tensor categories, in: Lie Theory and Geometry, in honor of Bertram Kostant, Birkhäuser, Boston, 1993, Prog. Math. 123, 349-383 (1994; Zbl 0848.17031)]. For a nonzero complex number \(z\) and modules \(W_1\), \(W_2\) for a vertex operator algebra \(V\) the authors introduce the algebraic notion of \(P(z)\)-tensor product \(W_1 \boxtimes_{P(z)} W_2\). The rationality of \(V\) is sufficient for the construction of the \(P(z)\)-tensor product of two modules, but it is still not enough to guarantee the associativity of the \(P (\cdot\))-tensor product; among further assumptions one needs the convergence and extension property of products and iterates of intertwining operators for \(V\).
In the paper under review the author shows that for any \(V\), containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras, all needed conditions are satisfied and that the tensor product theory can be applied. In particular, for any pair \(p,q\) of relatively prime positive integers larger than 1, the category of minimal modules of central charge \(1-6 ((p-q)^2/pq)\) for the Virasoro algebra has a natural structure of a braided tensor category.