Reachable elements in basic classical Lie 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.