The Nilpotent Cone for Classical Lie Superalgebras

TL;DR

This paper introduces a new nilpotent cone concept for classical Lie superalgebras, proves finiteness of certain orbits, and relates it to the Duflo-Serganova commuting variety, extending previous work in the field.

Contribution

It defines the nilpotent cone for classical Lie superalgebras, proves finiteness of orbits, and links it to the Duflo-Serganova commuting variety, broadening understanding of superalgebra structures.

Findings

Finitely many orbits on the nilpotent cone for classical Lie superalgebras.

The Duflo-Serganova commuting variety is contained in the nilpotent cone.

Extension of previous results on commuting varieties to superalgebras.

Abstract

In this paper the authors introduce an analog of the nilpotent cone, ${\mathcal N}$, for a classical Lie superalgebra, ${\mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$ with $\text{Lie }G_{\bar{0}}={\mathfrak g}_{\bar{0}}$, it is shown that there are finitely many $G_{\bar{0}}$-orbits on ${\mathcal N}$. Later the authors prove that the Duflo-Serganova commuting variety, ${\mathcal X}$, is contained in ${\mathcal N}$ for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.