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El. knyga: Pseudo-reductive Groups

, (University of Michigan, Ann Arbor), (Stanford University, California)
  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 29-Jul-2010
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781139036641
Kitos knygos pagal šią temą:
Pseudo-reductive GroupsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: New Mathematical Monographs
  • Išleidimo metai: 29-Jul-2010
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781139036641
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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"--

Provided by publisher.

Recenzijos

'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 MATH

Daugiau informacijos

A comprehensive treatment of the theory. Includes numerous new results and a complete classification.
Introduction xi
Terminology, conventions, and notation xix
Part I Constructions, examples, and structure theory
1(146)
1 Overview of pseudo-reductivity
3(40)
1.1 Comparison with the reductive case
3(8)
1.2 Elementary properties of pseudo-reductive groups
11(5)
1.3 Preparations for the standard construction
16(9)
1.4 The standard construction and examples
25(9)
1.5 Main result
34(2)
1.6 Weil restriction and fields of definition
36(7)
2 Root groups and root systems
43(43)
2.1 Limits associated to 1-parameter subgroups
43(16)
2.2 Pseudo-parabolic subgroups
59(8)
2.3 Root groups in pseudo-reductive groups
67(11)
2.4 Representability of automorphism functors
78(8)
3 Basic structure theory
86(61)
3.1 Perfect normal subgroups of pseudo-reductive groups
86(8)
3.2 Root datum for pseudo-reductive groups
94(5)
3.3 Unipotent groups associated to semigroups of roots
99(15)
3.4 Bruhat decomposition and Levi subgroups
114(16)
3.5 Classification of pseudo-parabolic subgroups
130(17)
Part II Standard presentations and their applications
147(68)
4 Variation of (G1, k1 /k, T1, C)
149(13)
4.1 Absolutely simple and simply connected fibers
149(5)
4.2 Uniqueness of (G1, k1 /k)
154(8)
5 Ubiquity of the standard construction
162(29)
5.1 Main theorem and central extensions
162(8)
5.2 Properties of standardness and standard presentations
170(9)
5.3 A standardness criterion
179(12)
6 Classification results
191(24)
6.1 The A1-case away from characteristic 2
192(6)
6.2 Types A2 and G2 away from characteristic 3
198(5)
6.3 General cases away from characteristics 2 and 3
203(12)
Part III General classification and applications
215(174)
7 The exotic constructions
217(38)
7.1 Calculations in characteristics 2 and 3
217(11)
7.2 Basic exotic pseudo-reductive groups
228(12)
7.3 Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups
240(15)
8 Preparations for classification in characteristics 2 and 3
255(24)
8.1 Further properties of basic exotic pseudo-reductive groups
255(5)
8.2 Exceptional and exotic pseudo-reductive groups
260(19)
9 The absolutely pseudo-simple groups in characteristic 2
279(63)
9.1 Type A1
280(10)
9.2 Root groups and birational group laws
290(9)
9.3 Construction of absolutely pseudo-simple groups with a non-reduced root system
299(19)
9.4 Classification of absolutely pseudo-simple groups with a non-reduced root system
318(24)
10 General case
342(16)
10.1 Factors with non-reduced root system and the generalized standard construction
342(9)
10.2 Classification via generalized standard groups
351(7)
11 Applications
358(31)
11.1 Maximal tori in pseudo-reductive groups
358(6)
11.2 Pseudo-semisimplicity
364(4)
11.3 Unirationality
368(4)
11.4 Structure of root groups and pseudo-parabolic subgroups
372(17)
Part IV Appendices
389(136)
A Background in linear algebraic groups
391(82)
A.1 Review of definitions
392(6)
A.2 Some results from the general theory
398(4)
A.3 Frobenius morphisms and non-affine groups
402(5)
A.4 Split reductive groups: Existence, Isomorphism, and Isogeny Theorems
407(15)
A.5 Weil restriction generalities
422(19)
A.6 Groups without Levi subgroups
441(5)
A.7 Lie algebras and Weil restriction
446(11)
A.8 Lie algebras and groups of multiplicative type
457(16)
B Tits' work on unipotent groups in nonzero characteristic
473(21)
B.1 Subgroups of vector groups
473(6)
B.2 Wound unipotent groups
479(3)
B.3 The cckp-kernel
482(3)
B.4 Torus actions on unipotent groups
485(9)
C Rational conjugacy in connected groups
494(31)
C.1 Pseudo-completeness
494(10)
C.2 Conjugacy results in the smooth affine case
504(7)
C.3 Split unipotent subgroups of pseudo-reductive groups
511(8)
C.4 Beyond the smooth affine case
519(6)
References 525(2)
Index 527
Brian Conrad is a Professor in the Department of Mathematics at Stanford University. Ofer Gabber is Professor of Mathematics at the Institut des Hautes Études Scientifiques (IHÉS), France. Gopal Prasad is Raoul Bott Professor of Mathematics at the University of Michigan.
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