李代数和代数群 英文PDF电子书下载

1 Results on topological spaces 1

1.1 Irreducible sets and spaces 1

1.2 Dimension 4

1.3 Noetherian spaces 5

1.4 Constructible sets 6

1.5 Gluing topological spaces 8

2 Rings and modules 11

2.1 Ideals 11

2.2 Prime and maximal ideals 12

2.3 Rings of fractions and localization 13

2.4 Localizations of modules 17

2.5 Radical of an ideal 18

2.6 Local rings 19

2.7 Noetherian rings and modules 21

2.8 Derivations 24

2.9 Module of differentials 25

3 Integral extensions 31

3.1 Integral dependence 31

3.2 Integrally closed domains 33

3.3 Extensions of prime ideals 35

4 Factorial rings 39

4.1 Generalities 39

4.2 Unique factorization 41

4.3 Principal ideal domains and Euclidean domains 43

4.4 Polynomials and factorial rings 45

4.5 Symmetric polynomials 48

4.6 Resultant and discriminant 50

5 Field extensions 55

5.1 Extensions 55

5.2 Algebraic and transcendental elements 56

5.3 Algebraic extensions 56

5.4 Transcendence basis 58

5.5 Norm and trace 60

5.6 Theorem of the primitive element 62

5.7 Going Down Theorem 64

5.8 Fields and derivations 67

5.9 Conductor 70

6 Finitely generated algebras 75

6.1 Dimension 75

6.2 Noether's Normalization Theorem 76

6.3 Krull's Principal Ideal Theorem 81

6.4 Maximal ideals 82

6.5 Zariski topology 84

7 Gradings and filtrations 87

7.1 Graded rings and graded modules 87

7.2 Graded submodules 88

7.3 Applications 90

7.4 Filtrations 91

7.5 Grading associated to a filtration 92

8 Inductive limits 95

8.1 Generalities 95

8.2 Inductive systems of maps 96

8.3 Inductive systems of magmas,groups and rings 98

8.4 An example 100

8.5 Inductive systems of algebras 100

9 Sheaves of functions 103

9.1 Sheaves 103

9.2 Morphisms 104

9.3 Sheaf associated to a presheaf 106

9.4 Gluing 109

9.5 Ringed space 110

10 Jordan decomposition and some basic results on groups 113

10.1 Jordan decomposition 113

10.2 Generalities on groups 117

10.3 Commutators 118

10.4 Solvable groups 120

10.5 Nilpotent groups 121

10.6 Group actions 122

10.7 Generalities on representations 123

10.8 Examples 126

11 Algebraic sets 131

11.1 Affine algebraic sets 131

11.2 Zariski topology 132

11.3 Regular functions 133

11.4 Morphisms 134

11.5 Examples of morphisms 136

11.6 Abstract algebraic sets 138

11.7 Principal open subsets 140

11.8 Products of algebraic sets 142

12 Prevarieties and varieties 147

12.1 Structure sheaf 147

12.2 Algebraic prevarieties 149

12.3 Morphisms of prevarieties 151

12.4 Products of prevarieties 152

12.5 Algebraic varieties 155

12.6 Gluing 158

12.7 Rational functions 159

12.8 Local rings of a variety 162

13 Projective varieties 167

13.1 Projective spaces 167

13.2 Projective spaces and varieties 168

13.3 Cones and projective varieties 171

13.4 Complete varieties 176

13.5 Products 178

13.6 Grassmannian variety 180

14 Dimension 183

14.1 Dimension of varieties 183

14.2 Dimension and the number of equations 185

14.3 System of parameters 187

14.4 Counterexamples 190

15 Morphisms and dimension 191

15.1 Criterion of affineness 191

15.2 Affine morphisms 193

15.3 Finite morphisms 194

15.4 Factorization and applications 197

15.5 Dimension of fibres of a morphism 199

15.6 An example 203

16 Tangent spaces 205

16.1 A first approach 205

16.2 Zariski tangent space 207

16.3 Differential of a morphism 209

16.4 Some lemmas 213

16.5 Smooth points 215

17 Normal varieties 219

17.1 Normal varieties 219

17.2 Normalization 221

17.3 Products of normal varieties 223

17.4 Properties of normal varieties 225

18 Root systems 233

18.1 Reflections 233

18.2 Root systems 235

18.3 Root systems and bilinear forms 238

18.4 Passage to the field of real numbers 239

18.5 Relations between two roots 240

18.6 Examples of root systems 243

18.7 Base of a root system 244

18.8 Weyl chambers 247

18.9 Highest root 250

18.10 Closed subsets of roots 250

18.11 Weights 253

18.12 Graphs 255

18.13 Dynkin diagrams 256

18.14 Classification of root systems 259

19 Lie algebras 277

19.1 Generalities on Lie algebras 277

19.2 Representations 279

19.3 Nilpotent Lie algebras 282

19.4 Solvable Lie algebras 286

19.5 Radical and the largest nilpotent ideal 289

19.6 Nilpotent radical 291

19.7 Regular linear forms 292

19.8 Cartan subalgebras 294

20 Semisimple and reductive Lie algebras 299

20.1 Semisimple Lie algebras 299

20.2 Examples 301

20.3 Semisimplicity of representations 302

20.4 Semisimple and nilpotent elements 305

20.5 Reductive Lie algebras 307

20.6 Results on the structure of semisimple Lie algebras 310

20.7 Subalgebras of semisimple Lie algebras 313

20.8 Parabolic subalgebras 316

21 Algebraic groups 319

21.1 Generalities 319

21.2 Subgroups and morphisms 321

21.3 Connectedness 322

21.4 Actions of an algebraic group 325

21.5 Modules 326

21.6 Group closure 327

22 Affine algebraic groups 331

22.1 Translations of functions 331

22.2 Jordan decomposition 333

22.3 Unipotent groups 335

22.4 Characters and weights 338

22.5 Tori and diagonalizable groups 340

22.6 Groups of dimension one 345

23 Lie algebra of an algebraic group 347

23.1 An associative algebra 347

23.2 Lie algebras 348

23.3 Examples 352

23.4 Computing differentials 354

23.5 Adjoint representation 359

23.6 Jordan decomposition 362

24 Correspondence between groups and Lie algebras 365

24.1 Notations 365

24.2 An algebraic subgroup 365

24.3 Invariants 368

24.4 Functorial properties 372

24.5 Algebraic Lie subalgebras 375

24.6 A particular case 380

24.7 Examples 383

24.8 Algebraic adjoint group 383

25 Homogeneous spaces and quotients 387

25.1 Homogeneous spaces 387

25.2 Some remarks 389

25.3 Geometric quotients 391

25.4 Quotient by a subgroup 393

25.5 The case of finite groups 397

26 Solvable groups 401

26.1 Conjugacy classes 401

26.2 Actions of diagonalizable groups 405

26.3 Fixed points 406

26.4 Properties of solvable groups 407

26.5 Structure of solvable groups 409

27 Reductive groups 413

27.1 Radical and unipotent radical 413

27.2 Semisimple and reductive groups 415

27.3 Representations 416

27.4 Finiteness properties 420

27.5 Algebraic quotients 422

27.6 Characters 424

28 Borel subgroups,parabolic subgroups,Cartan subgroups 429

28.1 Borel subgroups 429

28.2 Theorems of density 432

28.3 Centralizers and tori 434

28.4 Properties of parabolic subgroups 435

28.5 Cartan subgroups 437

29 Cartan subalgebras,Borel subalgebras and parabolic subalgebras 441

29.1 Generalities 441

29.2 Cartan subalgebras 443

29.3 Applications to semisimple Lie algebras 446

29.4 Borel subalgebras 447

29.5 Properties of parabolic subalgebras 450

29.6 More on reductive Lie algebras 453

29.7 Other applications 454

29.8 Maximal subalgebras 456

30 Representations of semisimple Lie algebras 459

30.1 Enveloping algebra 459

30.2 Weights and primitive elements 461

30.3 Finite-dimensional modules 463

30.4 Verma modules 464

30.5 Results on existence and uniqueness 467

30.6 A property of the Weyl group 469

31 Symmetric invariants 471

31.1 Invariants of finite groups 471

31.2 Invariant polynomial functions 475

31.3 A free module 478

32 S-triples 481

32.1 Jacobson-Morosov Theorem 481

32.2 Some lemmas 484

32.3 Conjugation of S-triples 487

32.4 Characteristic 488

32.5 Regular and principal elements 489

33 Polarizations 493

33.1 Definition of polarizations 493

33.2 Polarizations in the semisimple case 494

33.3 A non-polarizable element 497

33.4 Polarizable elements 499

33.5 Richardson's Theorem 502

34 Results on orbits 507

34.1 Notations 507

34.2 Some lemmas 508

34.3 Generalities on orbits 509

34.4 Minimal nilpotent orbit 511

34.5 Subregular nilpotent orbit 513

34.6 Dimension of nilpotent orbits 517

34.7 Prehomogeneous spaces of parabolic type 518

35 Centralizers 521

35.1 Distinguished elements 521

35.2 Distinguished parabolic subalgebras 523

35.3 Double centralizers 525

35.4 Normalizers 528

35.5 A semisimple Lie subalgebra 530

35.6 Centralizers and regular elements 533

36 σ-root systems 537

36.1 Definition 537

36.2 Restricted root systems 539

36.3 Restriction of a root 544

37 Symmetric Lie algebras 549

37.1 Primary subspaces 549

37.2 Definition of symmetric Lie algebras 553

37.3 Natural subalgebras 554

37.4 Cartan subspaces 555

37.5 The case of reductive Lie algebras 557

37.6 Linear forms 559

38 Semisimple symmetric Lie algebras 561

38.1 Notations 561

38.2 Iwasawa decomposition 562

38.3 Coroots 565

38.4 Centralizers 568

38.5 S-triples 570

38.6 Orbits 573

38.7 Symmetric invariants 579

38.8 Double centralizers 584

38.9 Normalizers 588

38.10 Distinguished elements 589

39 Sheets of Lie algebras 593

39.1 Jordan classes 593

39.2 Topology of Jordan classes 596

39.3 Sheets 601

39.4 Dixmier sheets 603

39.5 Jordan classes in the symmetric case 605

39.6 Sheets in the symmetric case 608

40 Index and linear forms 611

40.1 Stable linear forms 611

40.2 Index of a representation 615

40.3 Some useful inequalities 616

40.4 Index and semi-direct products 618

40.5 Heisenberg algebras in semisimple Lie algebras 621

40.6 Index of Lie subalgebras ofBorel subalgebras 625

40.7 Seaweed Lie algebras 629

40 8 An upper bound for the index 630

40.9 Cases where the bound is exact 635

40.10 On the index of parabolic subalgebras 638

References 641

List of notations 645

Index 647