Reachable elements in basic classical Lie superalgebras

TL;DR

This paper investigates properties of nilpotent elements in basic classical Lie superalgebras, characterizing when they are reachable or satisfy the Panyushev property, and classifies such elements for various superalgebras.

Contribution

It extends the study of nilpotent elements from Lie algebras to Lie superalgebras, providing classifications and bases for centralizers in several cases.

Findings

e is reachable iff it satisfies the Panyushev property for certain superalgebras

Classification of reachable and strongly reachable elements in exceptional superalgebras

Provides bases for centralizers and their centers in specific superalgebras

Abstract

Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over \mathbb{C}, e\in\mathfrak{g}_{\bar{0}} a nilpotent element and \mathfrak{g}^{e} the centralizer of e in \mathfrak{g}. We study various properties of nilpotent elements in \mathfrak{g}, which have previously only been considered in the case of Lie algebras. In particular, we prove that e is reachable if and only if e satisfies the Panyushev property for \mathfrak{g}=\mathfrak{sl}(m|n), m\neq n or \mathfrak{psl}(n|n) and \mathfrak{osp}(m|2n). For exceptional Lie superalgebras \mathfrak{g}=D(2,1;\alpha), G(3), F(4), we give the classification of e which are reachable, strongly reachable or satisfy the Panyushev property. In addition, we give bases for \mathfrak{g}^{e} and its centre \mathfrak{z}(\mathfrak{g}^{e}) for \mathfrak{g}=\mathfrak{psl}(n|n), which completes results of Han on the relationship between \dim\mathfrak{g}^{e}, \dim\mathfrak{z}(\mathfrak{g}^{e}) and the labelled Dynkin diagrams for all basic classical Lie superalgebras.