Representations of the Fermion-Virasoro superalgebras

Abstract

Let $\delta =0$ or $\frac{1}{2}$. In this paper we introduce two infinite-dimensional Lie superalgebras: the Fermion superalgebra $F\left(\delta \right)$ and the Fermion-Virasoro superalgebra $S\left(\delta \right)$. We classify all simple smooth $F\left(\delta \right)$-modules, all simple weight $F\left(\delta \right)$-modules, all simple smooth $S\left(\delta \right)$-modules of nonzero level, and all simple Harish-Chandra $S\left(\delta \right)$-modules. We also classify all $S\left(\delta \right)$-modules structures on free $C\left[L_0\right]$-modules of rank 2 with non-trivial even and odd spaces (four different classes) where $L_0$ is a degree 0 element in the graded superalgebra $S\left(\delta \right)$.