幂结合代数中的半轴
Half-axes in power associative algebras
摘要：
设$A$是特征为$\ne 2$的域$\mathbb{F}$上的交换非结合代数. $A$中的半轴是一个幂等$e\in A$，使得$e$满足Jordan代数中的Peirce乘法规则，此外，${\rm ad}_e$的$1$ -本征空间（乘以$e$）是一维的。在本文中，我们认为身份$(*)$ $x^2x^2=x^4$和$x^3x^2=xx^4.$我们表明，如果身份$(*)$严格持有$A,$，然后得到（非常）有趣的身份之间的元素在本征空间的${\rm ad}_e$（注意，如果$|\mathbb{F}|>3$和身份$(*)$持有$A,$，然后他们持有严格的$A$）。证明了若$A$是Jordan型半本原轴代数（即，$A$是由半轴生成的），且恒等式$(*)$在$A,$中严格成立，则$A$是Jordan代数.
英文摘要：
Let $A$ be a commutative, non-associative algebra over a field $\mathbb{F}$ of characteristic $\ne 2$. A half-axis in $A$ is an idempotent $e\in A$ such that $e$ satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the $1$-eigenspace of ${\rm ad}_e$ (multiplication by $e$) is one dimensional. In this paper we consider the identities $(*)$ $x^2x^2=x^4$ and $x^3x^2=xx^4.$ We show that if identities $(*)$ hold strictly in $A,$ then one gets (very) interesting identities between elements in the eigenspaces of ${\rm ad}_e$ (note that if $|\mathbb{F}|>3$ and the identities $(*)$ hold in $A,$ then they hold strictly in $A$). Furthermore we prove that if $A$ is a primitive axial algebra of Jordan type half (i.e., $A$ is generated by half-axes), and the identities $(*)$ hold strictly in $A,$ then $A$ is a Jordan algebra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
