论文标题
替代$ M_2 $ -Algebras和$ {\MITγ} $ - 代数
Alternative $M_2$-algebras and ${\mit Γ}$-algebras
论文作者
论文摘要
最近，V.H.LópezSolís和I.Shestakov解决了N.Jacobson的旧问题，描述了包含$ 2 \ times 2 $矩阵代数$ M_2 $的Unital替代代数为Unital Subalgebra。在这里，我们通过6维替代超级级$ b（4,2）$和辅助$ \ mathbf {\ mit z} _2 $ graded algebra $ \mitγ$提供了另一个描述$ M_2 $ -Algebras。发生$ M_2 $ -Algebras的替代类别与$ \MITγ$ -Algebras类别是同构。对于任何关联和交换的代数$ a $，我们提供了$ {\mitγ} $ - 代数$ \mitγ（a）$的构造，这变成了Jordan Superalgebra；如果$ a $是一个域，则$ \mitγ（a）$是主要的超级级。我们还描述了免费的$ \mitγ$ - 代数并构建其基础。
Recently V.H.López Solís 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 $\mitΓ$. It occurs that the category of alternative $M_2$-algebras is isomorphic to the category of $\mitΓ$-algebras. For any associative and commutative algebra $A$, we give a construction of a ${\mit Γ}$-algebra $\mitΓ(A)$, which turns to be a Jordan superalgebra; if $A$ is a domain then $\mitΓ(A)$ is a prime superalgebra. We describe also the free $\mitΓ$-algebras and construct their bases.
