Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras

TL;DR

This paper investigates the existence of outer automorphisms in simple algebraic groups, resolves a specific open case for type ^3D_4, and introduces a Skolem-Noether theorem for Albert algebras, advancing understanding in algebraic group automorphisms.

Contribution

It proves the existence of outer automorphisms for the type ^3D_4 group and establishes a Skolem-Noether theorem for Albert algebras, addressing a key open problem.

Findings

Resolved the existence question for outer automorphisms in type ^3D_4 groups.

Developed a Skolem-Noether theorem for cubic étale subalgebras of Albert algebras.

Provided conditions for outer automorphisms of order 2 in simply connected groups of outer type A.

Abstract

The question of existence of outer automorphisms of a simple algebraic group $G$ arises naturally both when working with the Galois cohomology of $G$ and as an example of the algebro-geometric problem of determining which connected components of the automorphism group of $G$ have rational points. The existence question remains open only for four types of groups, and we settle one of the remaining cases, type $^3D_4$. The key to the proof is a Skolem-Noether theorem for cubic etale subalgebras of Albert algebras which is of independent interest. Necessary and sufficient conditions for a simply connected group of outer type $A$ to admit outer automorphisms of order 2 are also given.