Centres of centralizers of nilpotent elements in Lie superalgebras $\mathfrak{sl}(m|n)$ or $\mathfrak{osp}(m|2n)$

TL;DR

This paper characterizes the centralizers and their centers of nilpotent elements in Lie superalgebras associated with classical supergroups, providing explicit bases and relations to Dynkin diagrams.

Contribution

It offers explicit descriptions of centralizers and their centers in Lie superalgebras of types lgebra, including bases and Dynkin diagram relations, advancing understanding of their structure.

Findings

Bases for centralizers and their centers are constructed.

Relations between centers and Dynkin diagrams are established.

Structural descriptions of nilpotent element centralizers are provided.

Abstract

Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or $\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let $\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a superalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular, we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and $\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$. We also determine the labelled Dynkin diagram $\varDelta$ with respect to $e$ and subsequently describe the relation between $\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and $\varDelta$.