Pseudo-reductive Groups

This monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form.

Brian Conrad (Author), Ofer Gabber (Author), Gopal Prasad (Author)

9781107087231, Cambridge University Press

Hardback, published 4 June 2015

690 pages
20.6 x 15.8 x 4.8 cm, 1.18 kg

'[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)

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.

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.

Subject Areas: Number theory [PBH], Groups & group theory [PBG], Algebra [PBF], Mathematics [PB]