The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

TL;DR

This paper classifies complex 4-dimensional nilpotent noncommutative Jordan algebras algebraically and geometrically, identifying 18 distinct algebras and describing their geometric variety through Zariski closures of rigid and parametric families.

Contribution

It provides the first complete algebraic and geometric classification of 4-dimensional nilpotent noncommutative Jordan algebras.

Findings

18 non-isomorphic nilpotent noncommutative Jordan algebras identified

Geometric variety characterized by Zariski closures of rigid and parametric families

Classification advances understanding of algebraic structures in Jordan algebra theory

Abstract

We give algebraic and geometric classifications of complex $4$-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of $3$ rigid algebras and $2$ one-parametric families of algebras.