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Descent in Buildings (AM-190)$
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Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss

Print publication date: 2015

Print ISBN-13: 9780691166902

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691166902.001.0001

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Strictly Semi-linear Automorphisms

Strictly Semi-linear Automorphisms

Chapter:
(p.259) Chapter Thirty Strictly Semi-linear Automorphisms
Source:
Descent in Buildings (AM-190)
Author(s):

Bernhard M¨uhlherr

Holger P. Petersson

Richard M. Weiss

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691166902.003.0030

This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ‎ is a Moufang spherical building and Π‎ is the Coxeter diagram of Δ‎; here the chapter fixes an apartment Σ‎ of Δ‎ and a root α‎ of Σ‎. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.

Keywords:   strictly semi-linear automorphism, root, root group, Moufang spherical building, Coxeter diagram, apartment, Moufang set, anisotropic quadratic space, octonion division algebra, non-linear automorphism

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