目錄
PREFACE
Ⅰ.BASIC CONCEPTS
1.Definitions and first examples
1.1 The notion of Lie algebra
1.2 Linear Lie algebras
1.3 Lie algebras of derivations
1.4 Abstract Lie algebras
2.Ideals and homomorphisms
2.1 Ideals
2.2 Homomorphisms and representations
2.3 Automorphisms
3. Solvable and nilpotent Lie algebras
3.1 Solvability
3.2 Nilpotency
3.3 Proof of Engel's Theorem
Ⅱ.SEMISIMPLE LIE ALGEBRAS
4.Theorems of Lie and Cartan
4.1 Lie's Theorem
4.2 Jordan-Chevalley decomposition
4.3 Cartan's Criterion
5.Killing form
5.1 Criterion for semisimplicity
5.2 Simple ideals of L
5.3 Inner derivations
5.4 Abstract Jordan decomposition
6.Complete reducibifity of representations
6.1 Modules
6.2 Casimir element of a representation
6.3 Weyl's Theorem
6.4 Preservation of Jordan decomposition
7.Representations of sl (2,F)
7.1 Weights and maximal vectors
7.2 Classification of irreducible modules
8.Root space decomposition
8.1 Maximal toral subalgebras and roots
8.2 Centralizer of H
8.3 Orthogonality properties
8.4 Integrality properties
8.5 Rationality properties. Summary
Ⅲ.ROOT SYSTEMS
9.Axiomatics
9.1 Reflections in a euclidean space
9.2 Root systems
9.3 Examples
9.4 Pairs of roots
10.Simple roots and Weyl group
10.1 Bases and Weyl chambers
10.2 Lemmas on simple roots
10.3 The Weyl group
10.4 Irreducible root systems
11.Classification
11.1 Cartan matrix of
11.2 Coxeter graphs and Dynkin diagrams
11.3 Irreducible components
11.4 Classification theorem
12.Construction of root systems and automorphisms
12.1 Construction of types A-G
12.2 Automorphisms of
13.Abstract theory of weights
13.1 Weights
13.2 Dominant weights
13.3 The weight δ
13.4 Saturated sets of weights
Ⅳ.ISOMORPHISM AND CONJUGACY THEOREMS
14.Isomorphism theorem
14.1 Reduction to the simple case
14.2 Isomorphism theorem
14.3 Automorphisms
15.Cartan subalgebras
15.1 Decomposition of L relative to ad x
15.2 Engel subalgebras
15.3 Caftan subalgebras
15.4 Functorial properties
16.Conjugacy theorems
16.1 The group E(L)
16.2 Conjugacy of CSA's (solvable case)
16.3 Borel subalgebras
16.4 Conjugacy of Borel subalgebras
16.5 Automorphism groups
Ⅴ.EXISTENCE THEOREM
17.Universal enveloping algebras
17.1 Tensor and symmetric algebras
17.2 Construction of U(L)
17.3 PBW Theorem and consequences
17.4 Proof of PBW Theorem
17.5 Free Lie algebras
18.Generators and relations
18.1 Relations satisfied by L
18.2 Consequences of (S1)-($3)
18.3 Serre's Theorem
18.4 Application: Existence and uniqueness theorems
19.The simple algebras
19.1 Criterion for semisimplicity
19.2 The classical algebras
19.3 The algebra G2
Ⅵ.REPRESENTATION THEORY
20.Weights and maximal vectors
20.1 Weight spaces
20.2 Standard cyclic modules
20.3 Existence and uniqueness theorems
21.Finite dimensional modules
21.1 Necessary condition for finite dimension
21.2 Sufficient condition for finite dimension
21.3 Weight strings and weight diagrams
21.4 Generators and relations for V(λ)
22.Multiplicity formula
22.1 A universal Casimir element
22.2 Traces on weight spaces
22.3 Freudenthal's formula
22.4 Examples
22.5 Formal characters
23.Characters
23.1 Invariant polynomial functions
23.2 Standard cyclic modules and characters
23.3 Harish-Chandra's Theorem Appendix
24.Formulas of Weyl, Kostant, and Steinberg
24.1 Some functions on H
24.2 Kostant's multiplicity formula
24.3 Weyl's formulas
24.4 Steinberg's formula Appendix
Ⅶ.CHEVALLEY ALGEBRAS AND GROUPS
25.Chevalley basis of L
25.1 Pairs of roots
25.2 Existence of a Chevalley basis
25.3 Uniqueness questions
25.4 Reduction modulo a prime
25.5 Construction of Chevalley groups (adjoint type)
26.Kostant's Theorem
26.1 A combinatorial lemma
26.2 Special case: sl (2, F)
26.3 Lemmas on commutation
26.4 Proof of Kostant's Theorem
27.Admissible lattices
27.1 Existence of admissible lattices
27.2 Stabilizer of an admissible lattice
27.3 Variation of admissible lattice
27.4 Passage to an arbitrary field
27.5 Survey of related results
References
Afterword (1994)
Index of Terminology
Index of Symbols
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