Geometric classification of nilpotent Jordan algebras of dimension five

TL;DR

This paper classifies five-dimensional nilpotent Jordan algebras by analyzing the geometric structure of their algebraic variety, identifying five irreducible components and describing their orbit closures.

Contribution

It provides a complete geometric classification of five-dimensional nilpotent Jordan algebras, identifying the irreducible components and orbit structures.

Findings

The variety of five-dimensional nilpotent Jordan algebras has five irreducible components.

Four components correspond to closures of orbits of rigid algebras.

One component is the closure of an infinite family of non-rigid algebras.

Abstract

The variety $JorN_{5}$ of five-dimensional nilpotent Jordan algebras structures over an algebraically closed field is investigated. We show that $JorN_{5}$ is the union of five irreducible components, four of them correspond to the Zariski closure of the $GL_5$-orbits of four rigid algebras and the other one is the Zariski closure of an union of orbits of infinite family of algebras, none of them being rigid.