Degenerations of complex associative algebras of dimension three via Lie and Jordan algebras (Q6569343)

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 \(G=\mathrm{GL}(3,\mathbb C)\) on \(\boldsymbol\Lambda_3(\mathbb C)\) `by change of basis'. Authors 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.\N\NThe results are important. The paper is pretty technical. It is motivated by deformational quantization.