The algebraic and geometric classification of noncommutative Jordan superalgebras

TL;DR

This paper provides a comprehensive algebraic and geometric classification of complex 3-dimensional noncommutative Jordan superalgebras, including related superalgebras and their varieties, advancing the structural understanding of these algebraic systems.

Contribution

It offers the first complete classification of 3-dimensional noncommutative Jordan superalgebras and related structures, revealing new correspondences and classifications within algebraic and geometric frameworks.

Findings

Classified 3-dimensional noncommutative Jordan superalgebras algebraically and geometrically.

Established correspondences between superalgebras and related algebraic structures.

Provided classifications of various superalgebra varieties and subvarieties.

Abstract

The algebraic and geometric classifications of complex $3$-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of $3$-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $\mathfrak{CD}$-, $\mathfrak{s}_4$-, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative $\big($rigid$\big)$ superalgebras.