Abstract. We investigate the free field realization of the twisted Heisenberg-Virasoro algebra H at level zero. We completely describe the structure of the associated Fock representations. Using vertex-algebraic methods and screening operators we construct singular vectors in certain Verma modules as Schur polynomials. We completely solve the irreducibility problem for the tensor products of irreducible highest weight modules with intermediate series. We also determine the fusion rules for an interesting subcategory of H-modules. Finally, as an application we present a free-field realization of the W (2, 2)-algebra and interpret the W (2, 2)-singular vectors as H-singular vectors in Verma modules.

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.

Abstract. The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra H at level zero as modules for the W (2, 2)-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:14054322-4342, arXiv: .1707. We prove that the irreducible highest weight H-module is irreducible as W (2, 2)-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly W (2, 2) vertex algebra.

This paper is a continuation of [2]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg-Virasoro algebra H at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for H is presented by combining a bosonic construction of Whittaker modules from [7] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible selfdual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of H-modules.In Section 2 we recall from [2] a free field realization of H, the definition of vertex algebra Π(0) and its modules Π(p, r). We study an extension of the Heisenberg-Virasoro vertex algebra Π(0) ⊂ Π(0) and present a structure of Π(0)-modules Π(p, r) in Section 3. By using certain relation in Π(0)-modules we recover formula 1.2 in Section 4. Then we consider a deformed action of H on these modules and obtain a family of modules Π(p, r) [cf. Theorem 5.2, Theorem 5.5] with the following properties:• Π(p, r) is a logarithmic H-module with the following action of the element L(0) of the Virasoro algebra:

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.