Representations of the Fermion-Virasoro algebras

TL;DR

This paper introduces the Fermion and Fermion-Virasoro algebras, classifies their simple modules, and discovers four distinct module structures on certain free modules, advancing the understanding of these infinite-dimensional Lie superalgebras.

Contribution

It defines the Fermion and Fermion-Virasoro algebras and classifies all their simple modules, revealing four different module structures on free modules of rank 2.

Findings

Classified all simple smooth and weight modules of the algebras.

Identified all simple Harish-Chandra modules of the Fermion-Virasoro algebra.

Discovered four different module structures on free modules of rank 2.

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 $2$.