Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$

TL;DR

This paper characterizes when odd nilpotent elements in the Lie superalgebra rak{gl}(m|n) can be embedded into rak{osp}(1|2) subalgebras, extending classical embedding results to superalgebras.

Contribution

It establishes a criterion for embedding odd nilpotent elements into rak{osp}(1|2) subalgebras based on super Jordan matrices with odd-sized blocks.

Findings

Embedding occurs iff the element is in the orbit of a super Jordan matrix with only odd-sized blocks.

Defines super Jordan matrices for rak{gl}(m|n).

Provides necessary and sufficient conditions for the embedding.

Abstract

In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{gl}(m|n)$ lies inside an $\mathfrak{osp}(1|2)$-subalgebra. In the case of the classical Lie algebra $\mathfrak{gl}_m$, every nilpotent element can be embedded into an $\mathfrak{sl}_2$-subalgebra, which is the result of the Jacobson-Morozov Theorem. In the case of the Lie superalgebra $\mathfrak{gl}(m|n)$, we define super Jordan matrices and prove that an odd nilpotent element $e$ is contained in an $\mathfrak{osp}(1|2)$-subalgebra if and only if $e$ lies in the orbit of a super Jordan matrix consisting only of super Jordan blocks of odd size.