Subspaces of 7 x 7 skew-symmetric matrices related to the group G_2

TL;DR

This paper explores a special 7-dimensional subspace of skew-symmetric matrices linked to octonion algebras over a field, revealing its invariance under automorphisms and its rank properties depending on the algebra's nature.

Contribution

It identifies a unique invariant subspace of skew-symmetric matrices associated with octonion algebras and describes its rank characteristics in division and split cases.

Findings

Subspace is invariant under automorphism group of octonion algebra

Rank of elements varies with algebra type: 6 for division, 4 and 6 for split

Automorphism group is the exceptional group G_2(K) in the split case

Abstract

Let $K$ be a field of characteristic different from 2 and let $C$ be an octonion algebra over $K$. We show that there is a seven-dimensional subspace of $7\times 7$ skew-symmetric matrices over $K$ which is invariant under the automorphism group of $C$. This subspace consists of elements of rank 6 when $C$ is a division algebra, and elements of rank 4 and 6 when $C$ is a split algebra. In the latter case, the automorphism group is the exceptional group $G_2(K)$.