李代数和代数群 英文【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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1 Results on topological spaces1

1.1 Irreducible sets and spaces1

1.2 Dimension4

1.3 Noetherian spaces5

1.4 Constructible sets6

1.5 Gluing topological spaces8

2 Rings and modules11

2.1 Ideals11

2.2 Prime and maximal ideals12

2.3 Rings of fractions and localization13

2.4 Localizations of modules17

2.5 Radical of an ideal18

2.6 Local rings19

2.7 Noetherian rings and modules21

2.8 Derivations24

2.9 Module of differentials25

3 Integral extensions31

3.1 Integral dependence31

3.2 Integrally closed domains33

3.3 Extensions of prime ideals35

4 Factorial rings39

4.1 Generalities39

4.2 Unique factorization41

4.3 Principal ideal domains and Euclidean domains43

4.4 Polynomials and factorial rings45

4.5 Symmetric polynomials48

4.6 Resultant and discriminant50

5 Field extensions55

5.1 Extensions55

5.2 Algebraic and transcendental elements56

5.3 Algebraic extensions56

5.4 Transcendence basis58

5.5 Norm and trace60

5.6 Theorem of the primitive element62

5.7 Going Down Theorem64

5.8 Fields and derivations67

5.9 Conductor70

6 Finitely generated algebras75

6.1 Dimension75

6.2 Noether's Normalization Theorem76

6.3 Krull's Principal Ideal Theorem81

6.4 Maximal ideals82

6.5 Zariski topology84

7 Gradings and filtrations87

7.1 Graded rings and graded modules87

7.2 Graded submodules88

7.3 Applications90

7.4 Filtrations91

7.5 Grading associated to a filtration92

8 Inductive limits95

8.1 Generalities95

8.2 Inductive systems of maps96

8.3 Inductive systems of magmas,groups and rings98

8.4 An example100

8.5 Inductive systems of algebras100

9 Sheaves of functions103

9.1 Sheaves103

9.2 Morphisms104

9.3 Sheaf associated to a presheaf106

9.4 Gluing109

9.5 Ringed space110

10 Jordan decomposition and some basic results on groups113

10.1 Jordan decomposition113

10.2 Generalities on groups117

10.3 Commutators118

10.4 Solvable groups120

10.5 Nilpotent groups121

10.6 Group actions122

10.7 Generalities on representations123

10.8 Examples126

11 Algebraic sets131

11.1 Affine algebraic sets131

11.2 Zariski topology132

11.3 Regular functions133

11.4 Morphisms134

11.5 Examples of morphisms136

11.6 Abstract algebraic sets138

11.7 Principal open subsets140

11.8 Products of algebraic sets142

12 Prevarieties and varieties147

12.1 Structure sheaf147

12.2 Algebraic prevarieties149

12.3 Morphisms of prevarieties151

12.4 Products of prevarieties152

12.5 Algebraic varieties155

12.6 Gluing158

12.7 Rational functions159

12.8 Local rings of a variety162

13 Projective varieties167

13.1 Projective spaces167

13.2 Projective spaces and varieties168

13.3 Cones and projective varieties171

13.4 Complete varieties176

13.5 Products178

13.6 Grassmannian variety180

14 Dimension183

14.1 Dimension of varieties183

14.2 Dimension and the number of equations185

14.3 System of parameters187

14.4 Counterexamples190

15 Morphisms and dimension191

15.1 Criterion of affineness191

15.2 Affine morphisms193

15.3 Finite morphisms194

15.4 Factorization and applications197

15.5 Dimension of fibres of a morphism199

15.6 An example203

16 Tangent spaces205

16.1 A first approach205

16.2 Zariski tangent space207

16.3 Differential of a morphism209

16.4 Some lemmas213

16.5 Smooth points215

17 Normal varieties219

17.1 Normal varieties219

17.2 Normalization221

17.3 Products of normal varieties223

17.4 Properties of normal varieties225

18 Root systems233

18.1 Reflections233

18.2 Root systems235

18.3 Root systems and bilinear forms238

18.4 Passage to the field of real numbers239

18.5 Relations between two roots240

18.6 Examples of root systems243

18.7 Base of a root system244

18.8 Weyl chambers247

18.9 Highest root250

18.10 Closed subsets of roots250

18.11 Weights253

18.12 Graphs255

18.13 Dynkin diagrams256

18.14 Classification of root systems259

19 Lie algebras277

19.1 Generalities on Lie algebras277

19.2 Representations279

19.3 Nilpotent Lie algebras282

19.4 Solvable Lie algebras286

19.5 Radical and the largest nilpotent ideal289

19.6 Nilpotent radical291

19.7 Regular linear forms292

19.8 Cartan subalgebras294

20 Semisimple and reductive Lie algebras299

20.1 Semisimple Lie algebras299

20.2 Examples301

20.3 Semisimplicity of representations302

20.4 Semisimple and nilpotent elements305

20.5 Reductive Lie algebras307

20.6 Results on the structure of semisimple Lie algebras310

20.7 Subalgebras of semisimple Lie algebras313

20.8 Parabolic subalgebras316

21 Algebraic groups319

21.1 Generalities319

21.2 Subgroups and morphisms321

21.3 Connectedness322

21.4 Actions of an algebraic group325

21.5 Modules326

21.6 Group closure327

22 Affine algebraic groups331

22.1 Translations of functions331

22.2 Jordan decomposition333

22.3 Unipotent groups335

22.4 Characters and weights338

22.5 Tori and diagonalizable groups340

22.6 Groups of dimension one345

23 Lie algebra of an algebraic group347

23.1 An associative algebra347

23.2 Lie algebras348

23.3 Examples352

23.4 Computing differentials354

23.5 Adjoint representation359

23.6 Jordan decomposition362

24 Correspondence between groups and Lie algebras365

24.1 Notations365

24.2 An algebraic subgroup365

24.3 Invariants368

24.4 Functorial properties372

24.5 Algebraic Lie subalgebras375

24.6 A particular case380

24.7 Examples383

24.8 Algebraic adjoint group383

25 Homogeneous spaces and quotients387

25.1 Homogeneous spaces387

25.2 Some remarks389

25.3 Geometric quotients391

25.4 Quotient by a subgroup393

25.5 The case of finite groups397

26 Solvable groups401

26.1 Conjugacy classes401

26.2 Actions of diagonalizable groups405

26.3 Fixed points406

26.4 Properties of solvable groups407

26.5 Structure of solvable groups409

27 Reductive groups413

27.1 Radical and unipotent radical413

27.2 Semisimple and reductive groups415

27.3 Representations416

27.4 Finiteness properties420

27.5 Algebraic quotients422

27.6 Characters424

28 Borel subgroups,parabolic subgroups,Cartan subgroups429

28.1 Borel subgroups429

28.2 Theorems of density432

28.3 Centralizers and tori434

28.4 Properties of parabolic subgroups435

28.5 Cartan subgroups437

29 Cartan subalgebras,Borel subalgebras and parabolic subalgebras441

29.1 Generalities441

29.2 Cartan subalgebras443

29.3 Applications to semisimple Lie algebras446

29.4 Borel subalgebras447

29.5 Properties of parabolic subalgebras450

29.6 More on reductive Lie algebras453

29.7 Other applications454

29.8 Maximal subalgebras456

30 Representations of semisimple Lie algebras459

30.1 Enveloping algebra459

30.2 Weights and primitive elements461

30.3 Finite-dimensional modules463

30.4 Verma modules464

30.5 Results on existence and uniqueness467

30.6 A property of the Weyl group469

31 Symmetric invariants471

31.1 Invariants of finite groups471

31.2 Invariant polynomial functions475

31.3 A free module478

32 S-triples481

32.1 Jacobson-Morosov Theorem481

32.2 Some lemmas484

32.3 Conjugation of S-triples487

32.4 Characteristic488

32.5 Regular and principal elements489

33 Polarizations493

33.1 Definition of polarizations493

33.2 Polarizations in the semisimple case494

33.3 A non-polarizable element497

33.4 Polarizable elements499

33.5 Richardson's Theorem502

34 Results on orbits507

34.1 Notations507

34.2 Some lemmas508

34.3 Generalities on orbits509

34.4 Minimal nilpotent orbit511

34.5 Subregular nilpotent orbit513

34.6 Dimension of nilpotent orbits517

34.7 Prehomogeneous spaces of parabolic type518

35 Centralizers521

35.1 Distinguished elements521

35.2 Distinguished parabolic subalgebras523

35.3 Double centralizers525

35.4 Normalizers528

35.5 A semisimple Lie subalgebra530

35.6 Centralizers and regular elements533

36 σ-root systems537

36.1 Definition537

36.2 Restricted root systems539

36.3 Restriction of a root544

37 Symmetric Lie algebras549

37.1 Primary subspaces549

37.2 Definition of symmetric Lie algebras553

37.3 Natural subalgebras554

37.4 Cartan subspaces555

37.5 The case of reductive Lie algebras557

37.6 Linear forms559

38 Semisimple symmetric Lie algebras561

38.1 Notations561

38.2 Iwasawa decomposition562

38.3 Coroots565

38.4 Centralizers568

38.5 S-triples570

38.6 Orbits573

38.7 Symmetric invariants579

38.8 Double centralizers584

38.9 Normalizers588

38.10 Distinguished elements589

39 Sheets of Lie algebras593

39.1 Jordan classes593

39.2 Topology of Jordan classes596

39.3 Sheets601

39.4 Dixmier sheets603

39.5 Jordan classes in the symmetric case605

39.6 Sheets in the symmetric case608

40 Index and linear forms611

40.1 Stable linear forms611

40.2 Index of a representation615

40.3 Some useful inequalities616

40.4 Index and semi-direct products618

40.5 Heisenberg algebras in semisimple Lie algebras621

40.6 Index of Lie subalgebras ofBorel subalgebras625

40.7 Seaweed Lie algebras629

40 8 An upper bound for the index630

40.9 Cases where the bound is exact635

40.10 On the index of parabolic subalgebras638

References641

List of notations645

Index647