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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 V′α, β⊗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′α, β⊗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.

Virasoro algebra; Highest weight module; Intermediate series; Minimal model; Vertex operator algebra; Intertwining operator

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