Centers of centralizers of nilpotent elements in exceptional Lie superalgebras

TL;DR

This paper classifies the centralizers of nilpotent elements in certain exceptional Lie superalgebras, describing their structure, centers, and associated Dynkin diagrams, and relates these to group actions.

Contribution

It provides a detailed description of centralizers and their centers for nilpotent elements in exceptional Lie superalgebras of types D(2,1;α), G(3), and F(4), including Dynkin diagram labelling.

Findings

Explicit descriptions of centralizers and their centers.

Relations between center dimensions and Dynkin diagrams.

Classification of nilpotent orbits via labelled Dynkin diagrams.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and the labelled Dynkin diagram.