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Ⅰ.BASIC CONCEPTS 1

1. Definitions and first examples 1

1.1 The notion of Lie algebra 1

1.2 Linear Lie algebras 2

1.3 Lie algebras of derivations 4

1.4 Abstract Lie algebras 4

2. Ideals and homomorphisms 6

2.1 Ideals 6

2.2 Homomorphisms and representations 7

2.3 Automorphisms 8

3. Solvable and nilpotent Lie algebras 10

3.1 Solvability 10

3.2 Nilpotency 11

3.3 Proof of Engel’s Theorem 12

Ⅱ.SEMISIMPLE LIE ALGEBRAS 15

4. Theorems of Lie and Cartan 15

4.1 Lie’s Theorem 15

4.2 Jordan-Chevalley decomposition 17

4.3 Cartan’s Criterion 19

5. Killing form 21

5.1 Criterion for semisimplicity 21

5.2 Simple ideals of L 22

5.3 Inner derivations 23

5.4 Abstract Jordan decomposition 24

6. Complete reducibility of representations 25

6.1 Modules 25

6.2 Casimir element of a representation 27

6.3 Weyl’s Theorem 28

6.4 Preservation of Jordan decomposition 29

7. Representations of sI（2，F） 31

7.1 Weights and maximal vectors 31

7.2 Classification of irreducible modules 32

8. Root space decomposition 35

8.1 Maximal toral subalgebras and roots 35

8.2 Centralizer of H 36

8.3 Orthogonality properties 37

8.4 Integrality properties 38

8.5 Rationality properties.Summary 39

Ⅲ.ROOT SYSTEMS 42

9.Axiomatics 42

9.1 Reflections in a euclidean space 42

9.2 Root systems 42

9.3 Examples 43

9.4 Pairs of roots 44

10.Simple roots and Weyl group 47

10.1 Bases and Weyl chambers 47

10.2 Lemmas on simple roots 50

10.3 The Weyl group 51

10.4 Irreducible root systems 52

11.Classification 55

11.1 Cartan matrix of Φ 55

11.2 Coxeter graphs and Dynkin diagrams 56

11.3 Irreducible components 57

11.4 Classification theorem 57

12.Construction of root systems and automorphisms 63

12.1 Construction of types A-G 63

12.2 Automorphisms of Φ 65

13.Abstract theory of weights 67

13.1 Weights 67

13.2 Dominant weights 68

13.3 The weight 8 70

13.4 Saturated sets of weights 70

Ⅳ.ISOMORPHISM AND CONJUGACY THEOREMS 73

14.Isomorphism theorem 73

14.1 Reduction to the simple case 73

14.2 Isomorphism theorem 74

14.3 Automorphisms 76

15.Cartan subalgebras 78

15.1 Decomposition of L relative to ad x 78

15.2 Engel subalgebras 79

15.3 Cartan subalgebras 80

15.4 Functorial properties 81

16. Conjugacy theorems 81

16.1 The group ？（L） 82

16.2 Conjugacy of CSA’s（solvable case） 82

16.3 Borel subalgebras 83

16.4 Conjugacy of Borel subalgebras 84

16.5 Automorphism groups 87

Ⅴ.EXISTENCE THEOREM 89

17.Universal enveloping algebras 89

17.1 Tensor and symmetric algebras 89

17.2 Construction of ？（L） 90

17.3 PBW Theorem and consequences 91

17.4 Proof of PBW Theorem 93

17.5 Free Lie algebras 94

18. Generators and relations 95

18.1 Relations satisfied by L 96

18.2 Consequences of（S1）-（S3） 96

18.3 Serre’s Theorem 98

18.4 Application：Existence and uniqueness theorems 101

19. The simple algebras 102

19.1 Criterion for semisimplicity 102

19.2 The classical algebras 102

19.3 The algebra G2 103

Ⅵ.REPRESENTATION THEORY 107

20. Weights and maximal vectors 107

20.1 Weight spaces 107

20.2 Standard cyclic modules 108

20.3 Existence and uniqueness theorems 109

21. Finite dimensional modules 112

21.1 Necessary condition for finite dimension 112

21.2 Sufficient condition for finite dimension 113

21.3 Weight strings and weight diagrams 114

21.4 Generators and relations for V（λ） 115

22. Multiplicity formula 117

22.1 A universal Casimir element 118

22.2 Traces on weight spaces 119

22.3 Freudenthal’s formula 121

22.4 Examples 123

22.5 Formal characters 124

23. Characters 126

23.1 Invariant polynomial functions 126

23.2 Standard cyclic modules and characters 128

23.3 Harish-Chandra’s Theorem 130

Appendix 132

24. Formulas of Weyl，Kostant，and Steinberg 135

24.1 Some functions on H 135

24.2 Kostant’s multiplicity formula 136

24.3 Weyl’s formulas 138

24.4 Steinberg’s formula 140

Appendix 143

Ⅶ.CHEVALLEY ALGEBRAS AND GROUPS 145

25. Chevalley basis of L 145

25.1 Pairs of roots 145

25.2 Existence of a Chevalley basis 145

25.3 Uniqueness questions 146

25.4 Reduction modulo a prime 148

25.5 Construction of Chevalley groups（adjoint type） 149

26. Kostant’s Theorem 151

26.1 A combinatorial lemma 152

26.2 Special case：sl（2，F） 153

26.3 Lemmas on commutation 154

26.4 Proof of Kostant’s Theorem 156

27. Admissible lattices 157

27.1 Existence of admissible lattices 157

27.2 Stabilizer of an admissible lattice 159

27.3 Variation of admissible lattice 161

27.4 Passage to an arbitrary field 162

27.5 Survey of related results 163

References 165

Index of Terminology 167

Index of Symbols 170