Derivations of octonion matrix algebras

TL;DR

This paper computes the derivation algebras of hermitian and antihermitian matrices over octonion algebras across all dimensions, extending the known connections between octonions and exceptional Lie algebras.

Contribution

It generalizes the derivation algebra computations for matrices over octonions to all dimensions, broadening the understanding of octonion-related algebraic structures.

Findings

Derived the structure of derivation algebras for hermitian matrices over octonions.

Derived the structure of derivation algebras for antihermitian matrices over octonions.

Extended known results from 3x3 matrices to all matrix dimensions.

Abstract

It is well-known that the exceptional Lie algebras $\mathfrak{f}_4$ and $\mathfrak{g}_2$ arise from the octonions as the derivation algebras of the $3\times3$ hermitian and $1\times1$ antihermitian matrices, respectively. Inspired by this, we compute the derivation algebras of the spaces of hermitian and antihermitian matrices over an octonion algebra in all dimensions.