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

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 $S=S_{\bar{0}}\oplus S_{\bar{1}}$ 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.