Irreducible Components of the Varieties of Jordan Superalgebras of Types $(1,3)$ and $(3,1)$

TL;DR

This paper analyzes the geometry of the four-dimensional Jordan superalgebras for types $(1,3)$ and $(3,1)$ over an algebraically closed field of characteristic $0$, recasting the problem as the study of varieties $\mathcal{JS}^{(m,n)}$ and their irreducible components via $G$-orbits and Zariski closures. It establishes that the $(1,3)$-type variety decomposes into the closures of the orbits of $11$ rigid superalgebras, while the $(3,1)$-type variety decomposes into the closures of the orbits of $21$ rigid superalgebras, providing explicit descriptions of these components. A notable result is the existence of a four-dimensional solvable rigid Jordan superalgebra, which shows that an analogue of the Vergne conjecture fails in the superalgebra setting. The work further yields a geometric classification for associative and supercommutative four-dimensional algebras of these types and details the structure of nilpotent subvarieties, offering a groundwork for complete four-dimensional classification including the yet-unresolved type $(2,2)$ case in a follow-up study.

Abstract

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras, respectively. In both cases, the irreducible components of the varieties are described. Furthermore, we exhibit a four-dimensional solvable rigid Jordan superalgebra, showing that an analogue to the Vergne conjecture for Jordan superalgebras does not hold.

Figures (2)

• Figure 1: Hasse diagram of deformations for Jordan superalgebras of type $(1,3)$
• Figure 2: Hasse diagram of deformations for Jordan superalgebras of type $(3,1)$

Theorems & Definitions (12)

• Lemma 1
• Remark 2
• Theorem 3
• Corollary 4
• Corollary 5
• Lemma 6
• proof
• Theorem 7
• proof
• Remark 8