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

2022/12/20 by Ivanova, N. M., Pallikaros, C. A. · 1 citation
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2212.10635

Abstract

Let \boldsymbolΛ₃(\mathbb C) (=\mathbb C²⁷) 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 \boldsymbolΛ₃(\mathbb C) `by change of basis'. We determine the complete degeneration picture inside the algebraic subset \mathcal A^s₃ of \boldsymbolΛ₃(\mathbb C) consisting of associative algebra structures via the corresponding information on the algebraic subsets \mathcal L₃ and \mathcal J₃ of \boldsymbolΛ₃(\mathbb C) of Lie and Jordan algebra structures respectively. This is achieved with the help of certain G-module endomorphisms ϕ₁, ϕ₂ of \boldsymbolΛ₃(\mathbb C) which map \mathcal A^s₃ onto algebraic subsets of \mathcal L₃ and \mathcal J₃ respectively.

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