Two classes of non-weight modules over the twisted Heisenberg–Virasoro algebra

Abstract

In the present paper, we construct two classes of non-weight modules Ω(λ, α, β) ⊗ Ind(M ) and M V, Ω(λ, α, β) over the twisted Heisenberg-Virasoro algebra, which are both associated with the modules Ω (λ, α, β). We present the necessary and sufficient conditions under which modules in these two classes are irreducible and isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules Ind y,λ (C RS ) and Ind z,λ (C P Q ) over the twisted Heisenberg-Virasoro algebra and then apply the established results to give irreducible conditions for Ind y,λ (C RS ) and Ind z,λ (C P Q ).[L m , I n ] = nI m+n + δ m+n,0 (m 2 + m)C 2 ,Clearly, the subspaces spanned by {I m , C 3 | 0 = m ∈ Z} and by {L m , C 1 | m ∈ Z} are respectively the Heisenberg algebra and the Virasoro algebra. Notice that the center of H is spanned by {C 0 := I 0 , C i | i = 1, 2, 3}. Moreover, the twisted Heisenberg-Virasoro algebra has a triangular decomposition:The twisted Heisenberg-Virasoro algebra is one of the most important Lie algebras both in mathematics and in mathematical physics, whose structure theory has extensively studied (see, e.g., [7,10,20]).A fundamental problem in the representation theory of the twisted Heisenberg-Virasoro algebra is to classify all its irreducible modules. In fact, the theory of weight modules with all weight subspaces being finite dimensional (namely, Harish-Chandra-modules) is welldeveloped. Irreducible weight modules over H with a nontrivial finite dimensional weight subspace were proved to be Harish-Chandra modules [22]. And irreducible Harish-Chandra H-modules were classified in [15], each of which was shown to be either the highest (or lowest) weight module, or the module of intermediate series, consistent with the well-known result for Virasoro algebra [16]. While weight modules with an infinite dimensional weight subspace were also studied (see [6,19]).Non-weight modules constitute the other important ingredients of the representation theory of H, the study of which is definitely necessary and became popular in the last few years. A large class of new non-weight irreducible H-modules were constructed in [3], which includes the highest weight modules and Whittaker modules. Non-weight H-modules whose restriction to the universal enveloping algebra of the degree-0 part (modulo center) are free of rank 1 were studied in [4] (see also [8]). While by twisting the weight modules, the authors [6] obtained a family of new non-weight irreducible H-modules. However, the theory of non-weight H-modules is far more from being well-developed.As a continuation of [6], we still study the representation theory of H in this paper. But we shall be concerned with non-weight H-modules. To be more precisely, we construct two classes of new irreducible non-weight H-modules in the present paper, which are both related to the modules Ω(λ, α, β) (see [4]). It is well known that an important way of constructing modules is to consider the linear tensor product of two modules,...

Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules

“…Some of restricted D-modules with zero level are constructed in [6], and Some of restricted D-modules with nonzero level are constructed in [14]. There are other classes of non-weight modules over D, see [7,8,19], where a class of modules in [8] are a sepcial class of modules in [19].…”

Irreducible tensor product modules over the affine-Virasoro algebra of type $A_1$

“…It is well known that an important way to construct new modules over an algebra is to consider the linear tensor product of two known modules over the algebra (cf. [5,6], [11], [23], [26]). The purpose of the present paper is to construct new irreducible non-weight L-modules by taking tensor product of irreducible modules defined in [7] with irreducible highest weight modules.…”

Irreducible Twisted Heisenberg–Virasoro Modules from Tensor Products

“…We note that M V, Ω(λ, α, β) is reducible if and only if V ∼ = V α,δ d,0 τ for some τ ∈ C such that δ d,0 βτ = 0 (see [5]). Proof.…”

“…While weight modules with an infinite dimensional weight spaces were also studied (see [4,14]). In the last few years, various families of non-weight irreducible twisted Heisenberg-Virasoro modules were investigated (see, e.g., [2][3][4][5][6]10]). These are basically various versions of Whittaker modules and U(CL 0 )-free modules constructed using different tricks.…”