Simple Harish-Chandra modules over super Schrödinger algebra in (1+1) dimensional spacetime

Abstract

The N = 1 super Schrödinger algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}(1|1)$\end{document}S(1|1) in (1+1) dimensional spacetime contains a subalgebra isomorphic to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(1|2)$\end{document}osp(1|2)-module. Let V be a simple weight module for the N = 1 super Schrödinger algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}(1|1)$\end{document}S(1|1) but not a simple \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(1|2)$\end{document}osp(1|2)-module. Let ω ∈ supp(V). If V is neither a highest weight module nor a lowest weight module for \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}(1|1)$\end{document}S(1|1), we prove that \documentclass[12pt]{minimal}\begin{document}$\mathrm{supp}(V)=\omega +\mathbb {Z}$\end{document} supp (V)=ω+Z, and all non-trivial weight spaces of V have the same dimension. We prove that if V is a Harish-Chandra module, then it is a highest weight module, or lowest weight module, or a twisted localization of a highest weight module.

Simple Harish-Chandra supermodules over the super Schrödinger algebra

“…x : U (q) → U (q) and the assignment Zhang et al [21] proved the following proposition. Let V be a simple weight S-supermodule with finite dimensional weight spaces.…”

“…First of all, we know that B (u) x (N (λ, c)) ∈ N (see [21,Proposition 5.2]). Now assume that B (N (λ , c )).…”

“…First, we recall the twisting functor defined in [21]. Let u ∈ {q, f }, denote U (u) the corresponding localization of U = U(S).…”

“…The N = 1 super Schrödinger algebra in (1 + 1)-dimensional spacetime is defined to be Lie superalgebra: S = S(1 | 1) = S0 ⊕ S1, where the even part S0 = span C {e, f, h, p, q, z} is the Schrödinger algebra: Aizawa [1,2] investigated the Verma modules of super Schrödinger algebras in low dimensional spacetime. And other researchers in [21] studied the simple weight modules over S. However, it seems that the supermodules (A left S-supermodules is a supervector space V which is a left S-module in the usual sense such that S θ V τ ⊆ V θ+τ for θ, τ ∈ Z/2Z ) for this Lie superalgebra are not studied well.…”