Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra

TL;DR

This paper introduces $ abla$-operators for the Ovsienko-Roger superalgebra, enabling the classification of all simple cuspidal modules and related Harish-Chandra modules for various Lie superalgebras.

Contribution

It develops $ abla$-operators for the Ovsienko-Roger superalgebra and classifies all simple cuspidal modules, extending to related Lie superalgebras.

Findings

Classified all simple cuspidal modules for Ovsienko-Roger superalgebras.

Extended classification to related Lie superalgebras like N=1 BMS3 and super W(2,2).

Provided a framework for understanding modules over superalgebras using $ abla$-operators.

Abstract

With the $\Omega$-operators for the Virasoro algebra \cite{BF} and the super Virasoro algebra in \cite{CL, CLL}, we get the $\Omega$-operators for the Ovsienko-Roger superalgebras in this paper and then use it to classify all simple cuspidal modules for the $\bZ$-graded and $\frac12\bZ$-graded Ovsienko-Roger superalgebras. By this result, we can easily classify all simple Harish-Chandra modules over some related Lie superalgebras, including the $N=1$ BMS$_3$ algebra, the super $W(2,2)$, etc.