The algebraic and geometric classification of noncommutative Jordan algebras

TL;DR

This paper introduces a method to classify noncommutative Jordan algebras algebraically and geometrically, providing classifications for specific dimensions and types, and analyzing their algebraic varieties.

Contribution

It develops a novel method linking Jordan algebra classification to noncommutative cases and applies it to classify 3- and 4-dimensional algebras, including geometric aspects.

Findings

Classified complex 3-dimensional noncommutative Jordan algebras

Classified 3-dimensional Kokoris, Poisson, and Poisson-Jordan algebras

Described irreducible components of algebraic varieties

Abstract

In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex $3$-dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex $3$-dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex $4$-dimensional nilpotent Kokoris and standard algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components.