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

Abstract

We show that subsingular vectors may exist in Verma modules over W (2, 2), and present the subquotient structure of these modules. We prove conditions for irreducibility of the tensor product of intermediate series module with a highest weight module. Relation to intertwining operators over vertex operator algebra associated to W (2, 2) is discussed. Also, we study the tensor product of intermediate series and a highest weight module over the twisted Heisenberg-Virasoro algebra, and present series of irreducible modules with infinite-dimensional weight spaces.

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.…”

“…The proof of (1) uses the fact that for α = α 0 we have Φ 0 (Λ) = 0 and Φ n (Λ) = 0 for n = 0. Then the assertion follows by using the arguments from [20,17].…”

“…is a H-module with infinite-dimensional weight spaces (see [20]). The tensor product V α,β,F ⊗L H (c L , 0, c L,I , h, h I ) is closely related with intertwining operators for vertex algebras.…”

“…Problem (A) was posed in [20] (see also [17]) but for the complete solution one needs explicit formulas for singular vectors. These formulas are obtained in the present paper, so in Theorems 5.2 and 5.5 we present the complete solution to problem (A).…”

Aspects of infinite dimensional ℓ-super Galilean conformal algebra

“…The algebra (1.1) is called W (2, 2) in the context of vertex operator algebras. The W (2, 2) algebra and its highest weight modules arise naturally in the studies of vertex operator algebra L( [41,42,43]. However, in the context of infinite dimensional GCA, the algebra (1.1) is a special case of a wider class of infinite dimensional Lie algebras.…”