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arXiv · 2212.01008

Alternative $M_2$-algebras and ${\mit \Gamma}$-algebras

Alexandre Grishkov·Ivan Shestakov
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Abstract

Recently V.H.L\'opez Sol\'is and I.Shestakov solved an old problem by N.Jacobson on describing of unital alternative algebras containing the $2\times 2$ matrix algebra $M_2$ as a unital subalgebra. Here we give another description of $M_2$-algebras via the 6-dimensional alternative superalgebra $B(4,2)$ and an auxiliar $\mathbf {\mit Z}_2$-graded algebra $\mit\Gamma$. It occurs that the category of alternative $M_2$-algebras is isomorphic to the category of $\mit\Gamma$-algebras. For any associative and commutative algebra $A$, we give a construction of a ${\mit \Gamma}$-algebra $\mit\Gamma(A)$, which turns to be a Jordan superalgebra; if $A$ is a domain then $\mit\Gamma(A)$ is a prime superalgebra. We describe also the free $\mit\Gamma$-algebras and construct their bases.

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Alexandre Grishkov, Ivan Shestakov. 2022-12-02. Alternative $M_2$-algebras and ${\mit \Gamma}$-algebras. https://arxiv.org/abs/2212.01008

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