Composition Algebras and Outer Automorphisms of Algebraic Groups

TL;DR

This paper establishes a categorical equivalence linking eight-dimensional composition algebras with quadratic form n to automorphism pairs of algebraic group schemes, extending prior symmetric algebra results.

Contribution

It introduces a new categorical equivalence connecting composition algebras and automorphisms of algebraic groups, generalizing previous symmetric algebra findings.

Findings

Categorical equivalence between composition algebras and automorphism pairs

Extension of results from symmetric to general composition algebras

Derivation of known composition algebra results from the new framework

Abstract

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action of the projective similarity group of $n$ on certain pairs of automorphisms of the group scheme $\mathbf{PGO}^+(n)$ defined over $k$. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.