The Variety of Jordan Superalgebras of dimension four and even part of dimension two

TL;DR

This paper classifies the variety of four-dimensional Jordan superalgebras with a two-dimensional even part, revealing 25 irreducible components and constructing a rigid superalgebra with non-vanishing second cohomology.

Contribution

It provides a detailed classification of the variety of such superalgebras and constructs a new example of a rigid superalgebra with non-zero second cohomology.

Findings

The variety has 25 irreducible components.

24 components correspond to orbit closures of rigid superalgebras.

Constructed a four-dimensional rigid superalgebra with non-vanishing second cohomology.

Abstract

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $2$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$. We prove that the variety has $25$ irreducible components, $24$ of them correspond to the Zariski closure of the $GL_2(\mathbb{F})\times GL_2(\mathbb{F})$-orbits of rigid superalgebras and the other one is the Zariski closure of an union of orbits of an infinite family of superalgebras, none of them being rigid. Furthermore, it is known that the question of the existence of a rigid Jordan algebra whose second cohomology group does not vanish is still an open problem. We solve this problem in the context of superalgebras, showing a four-dimensional rigid Jordan superalgebra whose second cohomology group does not vanish.