Composition algebras and outer automorphisms of algebraic groups via triality

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 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.

KEYWORDS:

• Affine group schemes
• composition algebras
• outer automorphisms
• triality

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A75
• 20G15
• 11E88

Notes

¹Despite this notation, we shall not assume associativity, commutativity or unitality when we use the word algebra in general.

²Unless otherwise stated, all group actions are assumed to be from the left.