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In the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudo-reductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

Linear algebraic groups. --- Group theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- "ient homomorphism. --- Cartan k-subgroup. --- Dynkin diagram. --- Isogeny Theorem. --- Isomorphism Theorem. --- Levi subgroup. --- SeveriЂrauer variety. --- Tits classification. --- Tits-style classification. --- Weil restriction. --- algebraic geometry. --- automorphism functor. --- automorphism scheme. --- automorphism. --- canonical central extensions. --- central "ient. --- central extension. --- characteristic 2. --- conformal isometry. --- degenerate quadratic form. --- double bond. --- exotic construction. --- field-theoretic invariant. --- generalized exotic group. --- generalized standard group. --- generalized standard presentation. --- generalized standard. --- isomorphism class. --- isomorphism. --- isotropic group. --- k-tame central extension. --- linear isomorphism. --- linear-algebraic invariant. --- maximal torus. --- minimal type. --- non-reduced root system. --- number theory. --- pseudo-isogeny. --- pseudo-reductive group. --- pseudo-semisimple group. --- pseudo-simple group. --- pseudo-simple k-group. --- pseudo-split form. --- pseudo-split. --- quadratic space. --- quadrics. --- rank-1. --- rank-2. --- rigidity property. --- root field. --- root system. --- scheme-theoretic center. --- semisimple "ient. --- semisimple k-group. --- structure theorem.

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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.

Linear algebraic groups. --- Group theory. --- 512.74 --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Linear algebraic groups

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Group schemes (Mathematics) --- Grothendieck, Alexandre, --- Demazure, Michel --- Grothendieck, Alexandre, - 1928-2014