Local and $2$-local automorphisms of Cayley algebras

Abstract

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 $\{\phi \in O(\mathcal{C},n)\mid \phi (1)=1\}.$ 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 $\text{char}\mathbb{F}\ne 2,3$. 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 $\{\phi \in O(\mathcal{C},n)\mid \phi (1)=1\}.$

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

References in corpus (1)

• Local and $2$-local derivations of Cayley algebras