Reachable elements in basic classical Lie superalgebras

Abstract

Let g=g0¯⊕g1¯ be a basic classical Lie superalgebra over C, e∈g0¯ a nilpotent element and ge the centralizer of e in g. We study various properties of nilpotent elements in 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 g=sl(m|n), m≠n or psl(n|n) and osp(m|2n). For exceptional Lie superalgebras g=D(2,1;α), 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 ge and its centre z(ge) for g=psl(n|n), which completes results of Han on the relationship between dim⁡ge, dim⁡z(ge) and the labelled Dynkin diagrams for all basic classical Lie superalgebras.