Pseudo-Reductive Groups

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. 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 MATHPreface 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.