Local and $2$-local derivations of Cayley algebras

TL;DR

This paper characterizes local and 2-local derivations on Cayley algebras, revealing their structure varies between split and division cases, and extends results to related Malcev algebras.

Contribution

It provides a complete description of local and 2-local derivations on Cayley algebras over arbitrary fields, highlighting differences between split and division cases, and applies findings to Malcev algebras.

Findings

Local derivations form a Lie algebra isomorphic to the orthogonal Lie algebra of trace-zero elements.

On split Cayley algebras, 2-local derivations are all derivations forming the Lie algebra g_2.

On division Cayley algebras, 2-local derivations coincide with local derivations, both isomorphic to the trace-zero subspace.

Abstract

The present paper is devoted to the description of local and $2$-local derivations on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $\mathfrak{n}$, let $\mathcal{C}_0$ be its subspace of trace $0$ elements. We prove that the space of all local derivations of $\mathcal{C}$ coincides with the Lie algebra $\{d\in (\mathcal{C},\mathfrak{n}) | d(1)=0\}$ which is isomorphic to the orthogonal Lie algebra $(\mathcal{C}_0,\mathfrak{n})$. Further we prove that, surprisingly, the behavior of $2$-local derivations depends on the Cayley algebra being split or division. Every $2$-local derivation on the split Cayley algebra is a derivation, i.e. they form the exceptional Lie algebra $\mathfrak{g}_2(\mathbb{F})$ if $\textrm{char}\mathbb{F}\neq 2,3$. On the other hand, on division Cayley algebras over a field $\mathbb{F}$, the sets of $2$-local derivations and local derivations coincide, and they are isomorphic to the Lie algebra $(\mathcal{C}_0,\mathfrak{n})$. As a corollary we obtain descriptions of local and $2$-local derivations of the seven dimensional simple non-Lie Malcev algebras over fields of characteristic $\neq 2,3$.