Representations of the Fermion-Virasoro algebras

Abstract

Let $\delta =0$ or $\frac{1}{2}$. In this paper, we introduce the Fermion algebra $F(\delta )$ and the Fermion-Virasoro algebra $\mathcal{S}(\delta )$. They are infinite-dimensional Lie superalgebras. All simple smooth $F(\delta )$-modules, all simple weight $F(\delta )$-modules, all simple smooth $\mathcal{S}(\delta )$-modules of nonzero level, and all simple Harish-Chandra $\mathcal{S}(\delta )$-modules are determined. Surprisingly, we have found four different $\mathcal{S}(\delta )$-module structures on free $\mathbb{C}[L_0]$-modules of rank $2$.