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

Let ${\mathbf{\Lambda }}_3(\mathbb{C})(={\mathbb{C}}^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb{C}$ considered as a $G$-module via the action of $G=\mathrm{G}\mathrm{L}(3,\mathbb{C})$ on ${\mathbf{\Lambda }}_3(\mathbb{C})$ `by change of basis’. We determine the complete degeneration picture inside the algebraic subset ${\mathcal{A}}_3^s$ of ${\mathbf{\Lambda }}_3(\mathbb{C})$ consisting of associative algebra structures via the corresponding information on the algebraic subsets ${\mathcal{L}}_3$ and ${\mathcal{J}}_3$ of ${\mathbf{\Lambda }}_3(\mathbb{C})$ of Lie and Jordan algebra structures respectively. This is achieved with the help of certain $G$-module endomorphisms ${\phi }_1$, ${\phi }_2$ of ${\mathbf{\Lambda }}_3(\mathbb{C})$ which map ${\mathcal{A}}_3^s$ onto algebraic subsets of ${\mathcal{L}}_3$ and ${\mathcal{J}}_3$ respectively.