Classification of finite-dimensional Lie superalgebras whose even part is a three-dimensional simple Lie algebra over a field of characteristic not two or three

Abstract

Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}={\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$ over $k$, where ${\mathfrak{g}}_0$ is a three-dimensional simple Lie algebra. If $\mathcal{Z}(\mathfrak{g})$ denotes the centre of $\mathfrak{g}$, the result is the following: either $\{{\mathfrak{g}}_1,{\mathfrak{g}}_1\}=\{0\}$ or ${\mathfrak{g}}_1=({\mathfrak{g}}_0\oplus k)\oplus \mathcal{Z}(\mathfrak{g})$ or $\mathfrak{g}\cong {\mathfrak{osp}}_k(1|2)\oplus \mathcal{Z}(\mathfrak{g})$.