Simple modules over the superconformal algebra ${\mathcal{S}}^{\prime }(1,n)$

Abstract

Let $n\ge 2$, and let $\mathcal{S}(1,n)$ be the Lie superalgebra of zero-superdivergence superderivations of $\mathbb{C}[t^{\pm 1}]\otimes \hat{I}›(n)$. Its derived algebra ${\mathcal{S}}^{\prime }(1,n):=[\mathcal{S}(1,n),\mathcal{S}(1,n)]$ is well known as a superconformal algebra. In this paper, we first study Shen-Larsson modules over ${\mathcal{S}}^{\prime }(1,n)$. These modules, introduced by G. Shen and T. A. Larsson, are constructed from modules over the Weyl superalgebra $K_{1,n}$ and the special linear Lie superalgebra $\mathfrak{sl}(1,n)$. We establish necessary and sufficient conditions for the simplicity of Shen-Larsson modules and investigate their simple subquotients in the non-simple case. Then as an application, building on the classification of simple cuspidal ${\mathcal{S}}^{\prime }(1,n)$-modules by C. Martínez, O. Mathieu and E. Zelmanov, we obtain an explicit construction of all simple cuspidal modules over ${\mathcal{S}}^{\prime }(1,n)$.