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

TL;DR

This paper introduces a new way to describe unital alternative algebras containing $M_2$ using a 6-dimensional superalgebra and a graded algebra, establishing an isomorphism between categories of these algebras.

Contribution

It provides a novel description of $M_2$-algebras via superalgebras and graded algebras, and constructs associated Jordan superalgebras with explicit bases.

Findings

Category of alternative $M_2$-algebras is isomorphic to $ ext{ extit{ extbf{ extit{ extbf{ extGamma}}}}}$-algebras.

Construction of $ ext{ extit{ extbf{ extGamma}}}(A)$ yields Jordan superalgebras from associative commutative algebras.

Explicit bases for free $ ext{ extit{ extbf{ extGamma}}}$-algebras are described.

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.