J.E.Humphreys(J.E. 汉弗雷斯),美国麻省大学(University of Massachusetts)数学系教授。
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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 Spe 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