The Variety of Jordan Superalgebras of dimension four and even part of dimension two
Isabel Hernández, María Eugenia Martin, Rodrigo Lucas Rodrigues
TL;DR
The paper addresses the geometric classification of Jordan superalgebras of total dimension $4$ with even part of dimension $2$ (type $(2,2)$). It uses deformation theory and the second cohomology $H^2$ to identify irreducible components of the variety ${\mathcal{JS}}^{(2,2)}$, showing it consists of $25$ components: $24$ are closures of orbits of rigid algebras and one is the closure of an infinite one-parameter family. A key result is the construction of a four-dimensional rigid Jordan superalgebra for which $H^2(\mathcal{J},\mathcal{J})\neq 0$, illustrating that rigidity does not imply vanishing $H^2$ in the superalgebra setting. The work also details the structure of associative and nilpotent subvarieties within ${\mathcal JS}^{(2,2)}$ and highlights open problems in the classification of $(2,2)$ Jordan superalgebras.
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.