《INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY》

Ⅰ.BASIC CONCEPTS1

1. Definitions and first examples1

1.1 The notion of Lie algebra1

1.2 Linear Lie algebras2

1.3 Lie algebras of derivations4

1.4 Abstract Lie algebras4

2. Ideals and homomorphisms6

2.1 Ideals6

2.2 Homomorphisms and representations7

2.3 Automorphisms8

3. Solvable and nilpotent Lie algebras10

3.1 Solvability10

3.2 Nilpotency11

3.3 Proof of Engel’s Theorem12

Ⅱ.SEMISIMPLE LIE ALGEBRAS15

4. Theorems of Lie and Cartan15

4.1 Lie’s Theorem15

4.2 Jordan-Chevalley decomposition17

4.3 Cartan’s Criterion19

5. Killing form21

5.1 Criterion for semisimplicity21

5.2 Simple ideals of L22

5.3 Inner derivations23

5.4 Abstract Jordan decomposition24

6. Complete reducibility of representations25

6.1 Modules25

6.2 Casimir element of a representation27

6.3 Weyl’s Theorem28

6.4 Preservation of Jordan decomposition29

7. Representations of sI（2，F）31

7.1 Weights and maximal vectors31

7.2 Classification of irreducible modules32

8. Root space decomposition35

8.1 Maximal toral subalgebras and roots35

8.2 Centralizer of H36

8.3 Orthogonality properties37

8.4 Integrality properties38

8.5 Rationality properties.Summary39

Ⅲ.ROOT SYSTEMS42

9.Axiomatics42

9.1 Reflections in a euclidean space42

9.2 Root systems42

9.3 Examples43

9.4 Pairs of roots44

10.Simple roots and Weyl group47

10.1 Bases and Weyl chambers47

10.2 Lemmas on simple roots50

10.3 The Weyl group51

10.4 Irreducible root systems52

11.Classification55

11.1 Cartan matrix of Φ55

11.2 Coxeter graphs and Dynkin diagrams56

11.3 Irreducible components57

11.4 Classification theorem57

12.Construction of root systems and automorphisms63

12.1 Construction of types A-G63

12.2 Automorphisms of Φ65

13.Abstract theory of weights67

13.1 Weights67

13.2 Dominant weights68

13.3 The weight 870

13.4 Saturated sets of weights70

Ⅳ.ISOMORPHISM AND CONJUGACY THEOREMS73

14.Isomorphism theorem73

14.1 Reduction to the simple case73

14.2 Isomorphism theorem74

14.3 Automorphisms76

15.Cartan subalgebras78

15.1 Decomposition of L relative to ad x78

15.2 Engel subalgebras79

15.3 Cartan subalgebras80

15.4 Functorial properties81

16. Conjugacy theorems81

16.1 The group ？（L）82

16.2 Conjugacy of CSA’s（solvable case）82

16.3 Borel subalgebras83

16.4 Conjugacy of Borel subalgebras84

16.5 Automorphism groups87

Ⅴ.EXISTENCE THEOREM89

17.Universal enveloping algebras89

17.1 Tensor and symmetric algebras89

17.2 Construction of ？（L）90

17.3 PBW Theorem and consequences91

17.4 Proof of PBW Theorem93

17.5 Free Lie algebras94

18. Generators and relations95

18.1 Relations satisfied by L96

18.2 Consequences of（S1）-（S3）96

18.3 Serre’s Theorem98

18.4 Application：Existence and uniqueness theorems101

19. The simple algebras102

19.1 Criterion for semisimplicity102

19.2 The classical algebras102

19.3 The algebra G2103

Ⅵ.REPRESENTATION THEORY107

20. Weights and maximal vectors107

20.1 Weight spaces107

20.2 Standard cyclic modules108

20.3 Existence and uniqueness theorems109

21. Finite dimensional modules112

21.1 Necessary condition for finite dimension112

21.2 Sufficient condition for finite dimension113

21.3 Weight strings and weight diagrams114

21.4 Generators and relations for V（λ）115

22. Multiplicity formula117

22.1 A universal Casimir element118

22.2 Traces on weight spaces119

22.3 Freudenthal’s formula121

22.4 Examples123

22.5 Formal characters124

23. Characters126

23.1 Invariant polynomial functions126

23.2 Standard cyclic modules and characters128

23.3 Harish-Chandra’s Theorem130

Appendix132

24. Formulas of Weyl，Kostant，and Steinberg135

24.1 Some functions on H135

24.2 Kostant’s multiplicity formula136

24.3 Weyl’s formulas138

24.4 Steinberg’s formula140

Appendix143

Ⅶ.CHEVALLEY ALGEBRAS AND GROUPS145

25. Chevalley basis of L145

25.1 Pairs of roots145

25.2 Existence of a Chevalley basis145

25.3 Uniqueness questions146

25.4 Reduction modulo a prime148

25.5 Construction of Chevalley groups（adjoint type）149

26. Kostant’s Theorem151

26.1 A combinatorial lemma152

26.2 Special case：sl（2，F）153

26.3 Lemmas on commutation154

26.4 Proof of Kostant’s Theorem156

27. Admissible lattices157

27.1 Existence of admissible lattices157

27.2 Stabilizer of an admissible lattice159

27.3 Variation of admissible lattice161

27.4 Passage to an arbitrary field162

27.5 Survey of related results163

References165

Index of Terminology167

Index of Symbols170