Degenerations of Jordan Algebras and ''Marginal'' Algebras

TL;DR

This paper classifies all degenerations of three-dimensional Jordan algebras over complex numbers, identifies irreducible components, and introduces the concept of 'marginal' Jordan algebras of level one across various algebraic structures.

Contribution

It provides a complete description of degenerations and irreducible components of 3D Jordan algebras and introduces 'marginal' algebras of level one in multiple algebraic contexts.

Findings

All degenerations of 3D Jordan algebras are described.

Irreducible components of the variety are classified.

'Marginal' algebras of level one are defined across various algebra types.

Abstract

We describe all degenerations of the variety $\mathfrak{Jord}_3$ of Jordan algebras of dimension three over $\mathbb{C}.$ In particular, we describe all irreducible components in $\mathfrak{Jord}_3.$ For every $n$ we define an $n$-dimensional rigid ''marginal'' Jordan algebra of level one. Also, we discuss ''marginal'' algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz, and anticommutative cases.