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Pseudo-reductive Groups

2nd Edition

Part of New Mathematical Monographs

  • Date Published: June 2015
  • availability: Available
  • format: Hardback
  • isbn: 9781107087231

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  • Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. In this second edition there is new material on relative root systems and Tits systems for general smooth affine groups, including the extension to quasi-reductive groups of famous simplicity results of Tits in the semisimple case. Chapter 9 has been completely rewritten to describe and classify pseudo-split absolutely pseudo-simple groups with a non-reduced root system over arbitrary fields of characteristic 2 via the useful new notion of 'minimal type' for pseudo-reductive groups. Researchers and graduate students working in related areas, such as algebraic geometry, algebraic group theory, or number theory will value this book, as it develops tools likely to be used in tackling other problems.

    • A strong collaboration of authors representing three important areas: number theory, algebraic geometry and algebraic groups
    • Presents foundational results very useful to mathematicians working in related areas
    • The second edition provides a complete classification of pseudo-reductive groups of minimal type
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    Reviews & endorsements

    Review of previous edition: 'This book is an impressive piece of work; many hard technical difficulties are overcome in order to provide the general structure of pseudo-reductive groups and to elucidate their classification by means of reasonable data. In view of the importance of this class of algebraic groups … and of the impact of a better understanding of them on the general theory of linear algebraic groups, this book can be considered a fundamental reference in the area.' Mathematical Reviews

    Review of previous edition: 'Researchers and graduate students working in related areas, such as algebraic geometry, algebraic group theory, or number theory will appreciate this book and find many deep ideas, results and technical tools that may be used in other branches of mathematics.' Zentralblatt MATH

    '[This book] is devoted to the elucidation of the structure and classification of pseudo-reductive groups over imperfect fields, completing the program initiated by J. Tits, A. Borel and T. Springer in the last three decades of the last century … [it] is a remarkable achievement and the definitive reference for pseudo-reductive groups. It certainly belongs in the library of anyone interested in algebraic groups and their arithmetic and geometry.' Felipe Zaldivar, MAA Reviews (maa.org/press/maa-reviews)

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    Product details

    • Edition: 2nd Edition
    • Date Published: June 2015
    • format: Hardback
    • isbn: 9781107087231
    • length: 690 pages
    • dimensions: 229 x 152 x 43 mm
    • weight: 1.18kg
    • availability: Available
  • Table of Contents

    Preface to the second edition
    Introduction
    Terminology, conventions, and notation
    Part I. Constructions, Examples, and Structure Theory:
    1. Overview of pseudo-reductivity
    2. Root groups and root systems
    3. Basic structure theory
    Part II. Standard Presentations and Their Applications:
    4. Variation of (G', k'/k, T', C)
    5. Ubiquity of the standard construction
    6. Classification results
    Part III. General Classification and Applications:
    7. The exotic constructions
    8. Preparations for classification in characteristics 2 and 3
    9. Absolutely pseudo-simple groups in characteristic 2
    10. General case
    11. Applications
    Part IV. Appendices: A. Background in linear algebraic groups
    B. Tits' work on unipotent groups in nonzero characteristic
    C. Rational conjugacy in connected groups
    References
    Index.

  • Authors

    Brian Conrad, Stanford University, California
    Brian Conrad is a Professor in the Department of Mathematics at Stanford University.

    Ofer Gabber, Institut des Hautes Études Scientifiques, France
    Ofer Gabber is a Directeur de Recherches CNRS at the Institut des Hautes Études Scientifiques (IHÉS).

    Gopal Prasad, University of Michigan, Ann Arbor
    Gopal Prasad is Raoul Bott Professor of Mathematics at the University of Michigan.

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