Simple restricted modules over a new Lie superalgebra extended by the Ovsienko--Roger algebra

TL;DR

This paper introduces a new infinite-dimensional Lie superalgebra called the super extended Ovsienko--Roger algebra, constructs simple modules over it, and classifies certain Verma modules, revealing their reducibility.

Contribution

The paper defines a novel Lie superalgebra and classifies its simple modules and Verma modules, including their reducibility properties.

Findings

Constructed simple restricted modules induced from finite-dimensional solvable Lie superalgebras.

Classified simple generalized Verma modules over the new algebra.

Proved that all Verma modules over this algebra are reducible.

Abstract

In this paper, we introduce a new infinite-dimensional Lie superalgebra $\mathcal{S}$ called the super extended Ovsienko--Roger algebra. This algebra is obtained by determining the annihilation superalgebra of the Lie conformal superalgebra $S=S_{\bar0}\oplus S_{\bar{1}}$ with $S_{\bar{0}}=\mathbb{C}[\partial]L\oplus\mathbb{C}[\partial]W$, $S_{\bar{1}}=\mathbb{C}[\partial]G$ and non-trivial $\lambda$-brackets $[L_\lambda L]=(\partial+2\lambda)L$, $[L_\lambda G]=(\partial+\lambda)G$, $[L_\lambda W]=[G_\lambda G]=\partial W$. Then we construct a class of simple restricted $\mathcal{S}$-modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras under certain conditions. Moreover, we obtain the classification of simple generalized Verma modules over $\mathcal{S}$ and we show that the Verma module of $\mathcal{S}$ is always reducible.