Degenerations of complex associative algebras of dimension three via 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.