Local and $2$-local automorphisms of Cayley algebras

TL;DR

This paper characterizes local and 2-local automorphisms of Cayley algebras over any field, showing their structure depends on whether the algebra is split or division, with implications for automorphism groups.

Contribution

It provides a complete description of local and 2-local automorphisms of Cayley algebras, linking them to orthogonal groups and distinguishing cases based on algebra type.

Findings

Local automorphisms coincide with certain orthogonal transformations fixing 1.

On split Cayley algebras, 2-local automorphisms are automorphisms forming G2 group.

On division Cayley algebras, 2-local automorphisms and local automorphisms are the same.

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 $\{\varphi\in O(\mathcal{C},n)\mid \varphi(1)=1\}.$ 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 $\{\varphi\in O(\mathcal{C},n)\mid \varphi(1)=1\}.$