On Degenerations of Lie Superalgebras

TL;DR

This paper investigates the conditions under which one complex Lie superalgebra can degenerate into another, classifies all (2,2)-dimensional cases, and identifies rigid superalgebras including a nilpotent example.

Contribution

It provides necessary conditions for degenerations, a classification of (2,2)-dimensional Lie superalgebras, and discovers a nilpotent rigid superalgebra, expanding understanding of superalgebra structures.

Findings

The variety of (2,2)-dimensional Lie superalgebras has seven irreducible components.

Three components are closures of rigid superalgebras.

An example of a nilpotent rigid Lie superalgebra is found.

Abstract

We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that $\mathcal{LS}^{(2,2)}$ is the union of seven irreducible components, three of which are the Zariski closures of rigid Lie superalgebras. As byproduct, we obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the classical case where no example is known.