Degenerations of complex associative algebras of dimension three via Lie and Jordan algebras

TL;DR

This paper classifies all degenerations of 3-dimensional complex associative algebras by analyzing their relationships with Lie and Jordan algebra structures, using group actions and module endomorphisms.

Contribution

It provides a complete degeneration classification of 3D associative algebras through their connections with Lie and Jordan algebra structures, employing specific G-module endomorphisms.

Findings

Complete degeneration picture of 3D associative algebras

Identification of algebraic subsets related to Lie and Jordan structures

Use of G-module endomorphisms to map associative to Lie and Jordan algebras

Abstract

Let $\boldsymbol\Lambda_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbol\Lambda_3(\mathbb C)$ `by change of basis'. We determine the complete degeneration picture inside the algebraic subset $\mathcal A^s_3$ of $\boldsymbol\Lambda_3(\mathbb C)$ consisting of associative algebra structures via the corresponding information on the algebraic subsets $\mathcal L_3$ and $\mathcal J_3$ of $\boldsymbol\Lambda_3(\mathbb C)$ of Lie and Jordan algebra structures respectively. This is achieved with the help of certain $G$-module endomorphisms $\phi_1$, $\phi_2$ of $\boldsymbol\Lambda_3(\mathbb C)$ which map $\mathcal A^s_3$ onto algebraic subsets of $\mathcal L_3$ and $\mathcal J_3$ respectively.