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 ′ α,β ⊗ L(c, h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion from [Zh], 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 operator 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.

Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications

“…Similar irreducibility problems for the Virasoro algebra and the twisted Heisenberg-Virasoro algebra were studied in the papers [8,17,19,20,23]. Our irreducibility result is a certain refinement of these results.…”

Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and W(2, 2) algebra

“…so components v (n) span a module from intermediate series V ′ α,β,0 for β = 1−h 1 . Hence we have a nontrivial L-operator We shall prove this theorem in several steps, using analogous approach as in the Virasoro case (see [15]). First we show that some relation in L(c, h, h W ) is needed to obtain irreducibility.…”

“…We take nontrivial submodule U and x = v m−n ⊗ x 0 + · · · v m ⊗ x n ∈ U such that x i ∈ L(c, h, h I ) i , and show by induction on n that v k ⊗ v ∈ U . If L 1 x n = 0 or L 2 x n = 0 we follow the proof of Theorem 3.2 in [15]. Otherwise it must be I 1 x n = 0.…”

Tensor product weight modules over the Virasoro algebra

“…Our purpose in the present paper is to completely solve this problem. Partial solution to the same problem was independently given a little later in [23]. More precisely, we determine the necessary and sufficient conditions for such tensor products to be simple, using two polynomials obtained from the singular vectors of the original highest weight modules.…”