Pseudo-reductive Groups

Pseudo-reductive Groups

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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 self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, 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.show more

Product details

  • Electronic book text
  • CAMBRIDGE UNIVERSITY PRESS
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • 1139038966
  • 9781139038966

Review quote

'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 '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.' Zentralbaltt MATHshow more

Table of contents

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. Universality of the standard construction; 6. Classification results; Part III. General Classification and Applications: 7. General classification and applications; 8. Preparations for classification in characteristics 2 and 3; 9. The absolutely pseudo-simple case 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.show more