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

TL;DR

This paper classifies all finite-dimensional Lie superalgebras over a field of characteristic not two or three, with a three-dimensional simple Lie algebra as the even part, detailing their structures and isomorphism classes.

Contribution

It provides a complete classification of such Lie superalgebras, including explicit descriptions of their possible odd parts and centers, extending the understanding of their structure.

Findings

Classification includes cases with trivial and non-trivial odd parts.

Identifies isomorphism classes involving direct sums with the center.

Includes the specific case of the orthosymplectic superalgebra sp(1|2).

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 $\lbrace \mathfrak{g}_1,\mathfrak{g}_1 \rbrace=\lbrace 0 \rbrace$ or $\mathfrak{g}_1=(\mathfrak{g}_0 \oplus k)\oplus \mathcal{Z}(\mathfrak{g})$ or $\mathfrak{g}\cong \mathfrak{osp}_k(1|2)\oplus \mathcal{Z}(\mathfrak{g})$.