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

Abstract

In this paper, we discuss the structure of the tensor product $V_{\alpha,\beta }^{\prime}\otimes L(c,h)$ 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 $V_{\alpha ,\beta}^{\prime}\otimes L(c,h)$, 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 = − 22/5 and c = 1/2 is presented.