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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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Field-theoretic and linear-algebraic invariants

Field-theoretic and linear-algebraic invariants

Chapter:
(p.28) 3 Field-theoretic and linear-algebraic invariants
Source:
Classification of Pseudo-reductive Groups (AM-191)
Author(s):

Brian Conrad

Gopal Prasad

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691167923.003.0003

This chapter deals with field-theoretic and linear-algebraic invariants. It first presents a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2. It then considers an absolutely pseudo-simple group over a field k, along with a pseudo-split pseudo-reductive group over an arbitrary field k. It also establishes the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. This is followed by two examples that illustrate the root field in A1-cases. The chapter concludes with a discussion of a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, is supplemented with additional field-theoretic and linear-algebraic data.

Keywords:   field-theoretic invariant, linear-algebraic invariant, pseudo-split, characteristic 2, pseudo-simple group, pseudo-reductive group, semisimple quotient, root field, isomorphism class, minimal type

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