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Linear Algebraic Groups and Finite Groups of Lie Type [Kietas viršelis]

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(École Polytechnique Fédérale de Lausanne), (Technische Universität Kaiserslautern, Germany)
  • Formatas: Hardback, 324 pages, aukštis x plotis x storis: 231x155x20 mm, weight: 590 g, Worked examples or Exercises; 20 Tables, black and white; 6 Line drawings, unspecified
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 08-Sep-2011
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107008549
  • ISBN-13: 9781107008540
Kitos knygos pagal šią temą:
Linear Algebraic Groups and Finite Groups of Lie TypeDidesnis paveikslėlis
  • Formatas: Hardback, 324 pages, aukštis x plotis x storis: 231x155x20 mm, weight: 590 g, Worked examples or Exercises; 20 Tables, black and white; 6 Line drawings, unspecified
  • Serija: Cambridge Studies in Advanced Mathematics
  • Išleidimo metai: 08-Sep-2011
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1107008549
  • ISBN-13: 9781107008540
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781107008540.html
Raktažodžiai:
Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups, and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

This concise treatment covers many topics that are central to the subject, but missing from existing textbooks. It contains numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

Recenzijos

"This book provides a concise introduction to the theory of linear algebraic groups over an algebraically closed field (of arbitrary charachteristic) and the closely related finite groups of Lie type. Although there are several good books covering a similar range of topics, some important recent developments are treated here for the first time. This book is well written and the style of exposition is clear and reader-friendly, making it suitable for graduate students. The content is well organized, and the authors have sensibly avoided overloading the text with technical details." Timothy C. Burness for Mathematical Reviews

Daugiau informacijos

The first textbook on the subgroup structure, in particular maximal subgroups, for both algebraic and finite groups of Lie type.
Preface ix
List of tables
xiii
Notation xiv
PART I LINEAR ALGEBRAIC GROUPS
1(80)
1 Basic concepts
3(12)
1.1 Linear algebraic groups and morphisms
3(3)
1.2 Examples of algebraic groups
6(3)
1.3 Connectedness
9(4)
1.4 Dimension
13(2)
2 Jordan decomposition
15(5)
2.1 Decomposition of endomorphisms
15(3)
2.2 Unipotent groups
18(2)
3 Commutative linear algebraic groups
20(6)
3.1 Jordan decomposition of commutative groups
20(2)
3.2 Tori, characters and cocharacters
22(4)
4 Connected solvable groups
26(4)
4.1 The Lie-Kolchin theorem
26(1)
4.2 Structure of connected solvable groups
27(3)
5 G-spaces and quotients
30(6)
5.1 Actions of algebraic groups
30(3)
5.2 Existence of rational representations
33(3)
6 Borel subgroups
36(8)
6.1 The Borel fixed point theorem
36(3)
6.2 Properties of Borel subgroups
39(5)
7 The Lie algebra of a linear algebraic group
44(7)
7.1 Derivations and differentials
44(5)
7.2 The adjoint representation
49(2)
8 Structure of reductive groups
51(12)
8.1 Root space decomposition
51(2)
8.2 Semisimple groups of rank
53(4)
8.3 Structure of connected reductive groups
57(2)
8.4 Structure of semisimple groups
59(4)
9 The classification of semisimple algebraic groups
63(11)
9.1 Root systems
63(5)
9.2 The classification theorem of Chevalley
68(6)
10 Exercises for Part I
74(7)
PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
81(98)
11 BN-pairs and Bruhat decomposition
83(12)
11.1 On the structure of B
83(7)
11.2 Bruhat decomposition
90(5)
12 Structure of parabolic subgroups, I
95(9)
12.1 Parabolic subgroups
95(3)
12.2 Levi decomposition
98(6)
13 Subgroups of maximal rank
104(8)
13.1 Subsystem subgroups
104(3)
13.2 The algorithm of Borel and de Siebenthal
107(5)
14 Centralizers and conjugacy classes
112(9)
14.1 Semisimple elements
112(4)
14.2 Connectedness of centralizers
116(5)
15 Representations of algebraic groups
121(10)
15.1 Weight theory
121(4)
15.2 Irreducible highest weight modules
125(6)
16 Representation theory and maximal subgroups
131(9)
16.1 Dual modules and restrictions to Levi subgroups
131(3)
16.2 Steinberg's tensor product theorem
134(6)
17 Structure of parabolic subgroups, II
140(9)
17.1 Internal modules
140(5)
17.2 The theorem of Borel and Tits
145(4)
18 Maximal subgroups of classical type simple algebraic groups
149(17)
18.1 A reduction theorem
149(6)
18.2 Maximal subgroups of the classical algebraic groups
155(11)
19 Maximal subgroups of exceptional type algebraic groups
166(6)
19.1 Statement of the result
166(2)
19.2 Indications on the proof
168(4)
20 Exercises for Part II
172(7)
PART III FINITE GROUPS OF LIE TYPE
179(89)
21 Steinberg endomorphisms
181(7)
21.1 Endomorphisms of linear algebraic groups
181(3)
21.2 The theorem of Lang-Steinberg
184(4)
22 Classification of finite groups of Lie type
188(9)
22.1 Steinberg endomorphisms
188(5)
22.2 The finite groups GF
193(4)
23 Weyl group, root system and root subgroups
197(6)
23.1 The root system
197(3)
23.2 Root subgroups
200(3)
24 A BN-pair for GF
203(15)
24.1 Bruhat decomposition and the order formula
203(6)
24.2 BN-pair, simplicity and automorphisms
209(9)
25 Tori and Sylow subgroups
218(11)
25.1 F-stable tori
218(7)
25.2 Sylow subgroups
225(4)
26 Subgroups of maximal rank
229(7)
26.1 Parabolic subgroups and Levi subgroups
229(3)
26.2 Semisimple conjugacy classes
232(4)
27 Maximal subgroups of finite classical groups
236(8)
27.1 The theorem of Liebeck and Seitz
237(3)
27.2 The theorem of Aschbacher
240(4)
28 About the classes C1F,..., C7F and S
244(6)
28.1 Structure and maximally of groups in iF
244(2)
28.2 On the class S
246(4)
29 Exceptional groups of Lie type
250(13)
29.1 Maximal subgroups
250(4)
29.2 Lifting result
254(9)
30 Exercises for Part III
263(5)
Appendix A Root systems
268(14)
A.1 Bases and positive systems
268(4)
A.2 Decomposition of root systems
272(4)
A.3 The length function
276(2)
A.4 Parabolic subgroups
278(4)
Exercises
281(1)
Appendix B Subsystems
282(15)
B.1 The highest root
282(3)
B.2 The affine Weyl group
285(1)
B.3 Closed subsystems
286(4)
B.4 Other subsystems
290(2)
B.5 Bad primes and torsion primes
292(5)
Exercises
296(1)
Appendix C Automorphisms of root systems
297(3)
Exercises 300(1)
References 301(4)
Index 305
Gunter Malle is a Professor in the Department of Mathematics at the University of Kaiserslautern, Germany. Donna Testerman is a Lecturer in the Basic Sciences Faculty at the École Polytechnique Fédérale de Lausanne, Switzerland.