Contractions of low-dimensional nilpotent Jordan algebras

TL;DR

This paper classifies low-dimensional nilpotent Jordan algebras over complex numbers, describing their algebraic varieties, contractions, and deformations, and identifying irreducible components and rigid algebras.

Contribution

It provides a complete classification of three- and four-dimensional nilpotent Jordan algebras, including their geometric structure and deformation relations.

Findings

J2 and J3 are irreducible components.

J4 is the union of two Zariski closures of rigid algebras.

The paper extends contractions and deformations among these algebras.

Abstract

In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.