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Classification of Pseudo-reductive Groups (AM-191)$
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Brian Conrad and Gopal Prasad

Print publication date: 2015

Print ISBN-13: 9780691167923

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691167923.001.0001

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Automorphisms, isomorphisms, and Tits classification

Automorphisms, isomorphisms, and Tits classification

6 Automorphisms, isomorphisms, and Tits classification
Classification of Pseudo-reductive Groups (AM-191)

Brian Conrad

Gopal Prasad

Princeton University Press

This chapter considers automorphisms, isomorphisms, and Tits classification. It begins by establishing a version of the Isomorphism Theorem for pseudo-split pseudo-reductive groups, along with a pseudo-reductive variant of the Isogeny Theorem for split connected semisimple groups. The key to both proofs is a technique to construct pseudo-reductive subgroups of an ambient smooth affine group. Some instructive examples over imperfect fields k of characteristic 2 are given. The chapter goes on to discuss the behavior of the k-group ZG,C with respect to Weil restriction in the pseudoreductive case. It also describes automorphism schemes for pseudo-reductive groups, focusing only on the pseudo-semisimple case because commutative pseudo-reductive groups that are not tori generally have a non-representable automorphism functor. Finally, it examines Tits-style classification, using Dynkin diagrams to express the classification theorem.

Keywords:   automorphism, isomorphism, Isomorphism Theorem, pseudo-reductive group, Isogeny Theorem, Weil restriction, automorphism scheme, automorphism functor, Tits classification, Dynkin diagram

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