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

Skip Garibaldi; Holger P. Petersson

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arXiv · 1508.05696

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

Skip Garibaldi·Holger P. Petersson

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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.

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Skip Garibaldi, Holger P. Petersson. 2015-08-24. Outer automorphisms of algebraic groups and a Skolem-Noether theorem for Albert algebras. https://arxiv.org/abs/1508.05696

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