Classification of Simple Weight Modules for the Neveu–Schwarz Algebra with a Finite-Dimensional Weight Space

Abstract

We show that the support of a simple weight module over the Neveu-Schwarz algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite-dimensional. As a corollary we obtain that every simple weight module over the Neveu-Schwarz algebra, having a non-trivial finitedimensional weight space, is a Harish-Chandra module (and hence is either a highest or lowest weight module, or else a module of the intermediate series). This result generalizes a theorem which was originally given on the Virasoro algebra.

Irreducible Weight Modules with a Finite-Dimensional Weight Space over the Twisted N = 1 Schrödinger-Neveu-Schwarz Algebra

“…W = 0 and the given brackets, we obtain L n W = G r W = Y p W = M p W = 0 for all n = 1, q, q + 1, q + 2, · · · , r = 1 2 , q + 1 2 , q + 3 2 , · · · , p ∈ 1 2 Z + . We claim that G The following lemma can be easily verified, partially given in Lemma 2 of [11].…”

“…Z} is a nontrivial ideal of tsns, IV is a submodule of V for any tsns-module V , which gives the following lemma (such lemma was not used in [11]). For any irreducible weight tsns-module V , if we can prove IV = 0, then V will degenerate to be an irreducible weight ns-module.…”

“…Then U is simply a ns-module. Then we can use the same discussions as those given in Lemma 3 of [11], and have the following observations:…”

“…The following conjecture was given in [6]: The positive answers to the above question and conjecture were given in [7]. Such question and conjecture have already been solved on the truncated Virasoro algebras, the Walgebra W (2, 2), the twisted Schrödinger-Virasoro algebra, the twisted Heisenberg-Virasoro algebra and the Neveu-Schwarz algebra in [1,4,5,9,11]. In this note, we also give the positive answers to the above question and conjecture for the twisted N = 1 Schrödinger-Neveu-Schwarz algebra.…”

Classification of simple weight modules for super-Virasoro algebra with a finite-dimensional weight space

“…The Harish-Chandra modules over the super-Virasoro algebras were classified by Su [16]. The classification of the simple weight modules with a finite-dimensional weight space over the Neveu-Schwarz algebra was given in [20], and it played an important role in the study of tensor product weight representations of the Neveu-Schwarz algebra in [19].…”