Application of Vertex Algebras to the Structure Theory of Certain Representations Over the Virasoro Algebra

In this paper, we discuss the structure of the tensor product of an irreducible module from an intermediate series and irreducible highest-weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion from Zhang (J Algebra 190:1-10, 1997), and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operators for modules over vertex operator algebra yields reducibility of , which is a completely new point of view to this problem. As an example, the complete structure of the tensor product with minimal models c = -aEuro parts per thousand 22/5 and c = 1/2 is presented.