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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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Central extensions and groups locally of minimal type

Central extensions and groups locally of minimal type

Chapter:
(p.57) 4 Central extensions and groups locally of minimal type
Source:
Classification of Pseudo-reductive Groups (AM-191)
Author(s):

Brian Conrad

Gopal Prasad

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691167923.003.0004

This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar fact in the connected reductive case. The chapter then considers four phenomena that go beyond the quadratic case, along with a pseudo-reductive group of minimal type that is locally of minimal type. It shows that the pseudo-split absolutely pseudo-simple k-groups of minimal type with a non-reduced root system are classified over any imperfect field of characteristic 2. In this classification there is no effect if the “minimal type” hypothesis is relaxed to “locally of minimal type.”

Keywords:   central extension, minimal type, scheme-theoretic center, central quotient, pseudo-reductive group, pseudo-split, non-reduced root system, characteristic 2, pseudo-simple k-group

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