Whittaker modules for the super-Virasoro algebras

Abstract

In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.

A family of non-weight modules over the super-Virasoro algebras

“…for all r ∈ 1 2 + Z , it further yields that Ω R (λ, α) is a simple N S-module. 8 The following result gives an intimate connection between the two classes of modules constructed in Proposition 2.4 and Proposition 2.8.…”

“…In 1992, Mathieu [16] (also see [6,17]) classified all simple Harish-Chandra modules over the Virasoro algebra, i.e., simple weight modules with finite-dimensional weight spaces. Su [18] settled In [8], Whittaker modules over the super-Virasoro algebras are introduced, and necessary and sufficient conditions for irreducibility of these modules are given. The authors further studied the simple restricted modules over the super-Virasoro algebra of Neveu-Schwarz type in [9].…”

Simple restricted modules for Neveu-Schwarz algebra

“…All simple b (2) 0 -modules are constructed by Mazorchuk and Zhao in [30]. By [24], the Whittaker module W (ψ, ℓ) is simple if ψ is non-trivial, i.e., ψ(L 1 ) = 0 or ψ(L 2 ) = 0.…”

“…It is clear that p (0) = p. All finite dimensional simple modules over p (0) have been classified in [24]. Now we shall classify all finite-dimensional simple modules over p (t) for t ∈ Z + .…”

“…[12,15,16,19,31,34]). The authors [24] introduced Whittaker type modules over the super-Virasoro algebras and obtain necessary and sufficient conditions for irreducibility of these modules. The aforementioned results demonstrate that the Whittaker modules satisfy some properties that their non-super analogues do.…”