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