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.

On Four-Dimensional Unital Division Algebras over Fields of Characteristic not 2

“…if F is an equivalence of categories. 1 If (X, G, γ, F ) is a description of a groupoid Y and T ⊂ X is a transversal for the orbit set X/γ of the group action γ, then the subset F (T ) ⊂ Y is a transversal for the set Y / ≃ of all isoclasses of Y . Accordingly, Y is svelte (Subsection 2.1), and F (T ) classifies Y .…”

“…Apart from the groupoid C , the occurrence of this phenomenon has in fact been observed in various other contexts. As a sample, let us mention the groupoid of all 2-dimensional real division algebras [4, Theorem 3.3], the groupoid of all 4-dimensional absolute valued algebras [5,Proposition 5.3], and the groupoid of all 8-dimensional composition algebras over a field of characteristic not 2 [1,Corollary 4.6].…”