arXiv 2107.011472021-07-02math.RA

Cayley代数的局部和$2$ -局部自同构

Local and $2$-local automorphisms of Cayley algebras

Shavkat Ayupov, Alberto Elduque, Karimbergen Kudaybergenov

备注：: 9 pages. arXiv admin note: text overlap with arXiv:2105.08423

摘要：

本文研究了任意域$\mathbb{F}$上Cayley代数的局部自同构和2-局部自同构.给定一个范数为$n$的Cayley代数$\mathcal{C}$，设$O(\mathcal{C},n)$为相应的正交群.我们证明了$\mathcal{C}$的所有局部自同构的群与群$\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$重合。进一步证明了2-局部自同构的行为依赖于Cayley代数的分裂或除。分裂Cayley代数上的每一个2-局部自同构都是自同构，即它们构成例外李群$G_2(\mathbb{F})$，如果$\textrm{char}\mathbb{F}\neq 2,3$ .另一方面，在域$\mathbb{F}$上的除Cayley代数上，2-局部自同构群与局部自同构群重合，并且它们同构于群$\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$

英文摘要：

The present paper is devoted to the description of local and 2-local automorphisms on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $n$, let $O(\mathcal{C},n)$ be the corresponding orthogonal group. We prove that the group of all local automorphisms of $\mathcal{C}$ coincides with the group $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$ Further we prove that the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, i.e. they form the exceptional Lie group $G_2(\mathbb{F})$ if $\textrm{char}\mathbb{F}\neq 2,3$. On the other hand, on division Cayley algebras over a field $\mathbb{F}$, the groups of 2-local automorphisms and local automorphisms coincide, and they are isomorphic to the group $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$

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摘要：

英文摘要：

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