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Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether’s Normalization Theorem 6.3 Krull’s Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms and dimension 15.1 Criterion of affineness 15.2 Affine morphisms 15.3 Finite morphisms 15.4 Factorization and applications 15.5 Dimension of fibres of a morphism 15.6 An example 16. Tangent spaces 16.1 A first approach 16.2 Zariski tangent space 16.3 Differential of a morphism 16.4 Some lemmas 16.5 Smooth points 17. Normal varieties 17.1 Normal varieties 17.2 Normalization 17.3 Products of normal varieties 17.4 Properties of normal varieties 18. Root systems 18.1 Reflections 18.2 Root systems 18.3 Root systems and bilinear forms 18.4 Passage to the field of real numbers 18.5 Relation between two roots 18.6 Base of a root system 18.7 Weyl chambers 18.8 Highest root 18.9 Closed subsets of roots 18.10 Weights 18.11 Graphs 18.12 Dynkin diagrams 18.13 Classification of root systems 19. Lie algebras 19.1 Generalities on Lie algebras 19.2 Representations 19.3 Nilpotent Lie algebras 19.4 Solvable Lie algebras 19.5 Radical and the largest nilpotent ideal 19.6 Nilpotent radical 19.7 Regular linear forms 19.8 Cartan subalgebras 20. Semisimple and reductive Lie algebras 20.1 Semisimple Lie algebras 20.2 Examples 20.3 Semisimplicity of representations 20.4 Semisimple and nilpotent elements 20.5 Reductive Lie algebras 20.6 Results on the structure of semisimple Lie algebras 20.7 Subalgebras of semisimple Lie algebras 20.8 Parabolic subalgebras 21. Algebraic groups 21.1 Generalities 21.2 Subgroups and morphisms 21.3 Connectedness 21.4 Actions of an algebraic group 21.5 Modules 21.6 Group closure 22. Affine algebraic groups 22.1 Translations of functions 22.2 Jordan decomposition 22.3 Unipotent groups 22.4 Characters and weights 22.5 Tori and diagonalizable groups 22.6 Groups of dimension one 23. Lie algebra of an algebraic group 23.1 An associative algebra 23.2 Lie algebras 23.3 Examples 23.4 Computing differentials 23.5 Adjoint representation 23.6 Jordan decomposition 24. Correspondence between groups and Lie algebras 24.1 Notations 24.2 An algebraic subgroup 24.3 Invariants 24.4 Functorial properties 24.5 Algebraic Lie subalgebras 24.6 A particular case 24.7 Examples 24.8 Algebraic adjoint group 25. Homogeneous spaces and quotients 25.1 Homogeneous spaces 25.2 Some remarks 25.3 Geometric quotients 25.4 Quotient by a subgroup 25.5 The case of finite groups 26. Solvable groups 26.1 Conjugacy classes 26.2 Actions of diagonalizable groups 26.3 Fixed points 26.4 Properties of solvable groups 26.5 Structure of solvable groups 27. Reductive groups 27.1 Radical and unipotent radical 27.2 Semisimple and reductive groups 27.3 Representations 27.4 Finiteness properties 27.5 Algebraic quotients 27.6 Characters 28. Borel subgroups, parabolic subgroups and Cartan subgroups 28.1 Borel subgroups 28.2 Theorems of density 28.3 Centralizers and tori 28.4 Properties of parabolic subgroups 28.5 Cartan subgroups 29. Cartan subalgebras, Borel subalgebras and parabolic subalgebras 29.1 Generalities 29.2 Cartan subalgebras 29.3 Application to semisimple Lie algebras 29.4 Borel subalgebras 29.5 Properties of parabolic subalgebras 29.6 More on reductive Lie algebras 29.7 Other applications 29.8 Maximal subalgebras 30. Representations of semisimple Lie algebras 30.1 Enveloping algebra 30.2 Weights and primitive elements 30.3 Finite-dimensional modules 30.4 Verma modules 30.5 Results on existence and uniqueness 30.6 A property of the Weyl group 31. Symmetric invariants 31.1 Invariants of finite groups 31.2 Invariant polynomial functions 31.3 A free module 32. S-triples 32.1 Theorem of Jacobson-Morosov 32.2 Some lemmas 32.3 Conjugation of S-triples 32.4 Characteristic 32.5 Regular and principal elements 33. Polarizations 33.1 Definition of polarizations 33.2 Polarizations in the semisimple case 33.3 A non-polarizable element 33.4 Polarizable elements 33.5 Theorem of Richardson 34. Results on orbits 34.1 Notations 34.2 Some lemmas 34.3 Generalities on orbits 34.4 Minimal nilpotent orbit 34.5 Subregular nilpotent orbit 34.6 Dimension of nilpotent orbits 34.7 Prehomogeneous spaces of parabolic type 35. Centralizers 35.1 Distinguished elements 35.2 Distinguished parabolic subalgebras 35.3 Double centralizers 35.4 Normalizers 35.5 A semisimple Lie subalgebra 35.6 Centralizers and regular elements 36. s -root systems 36.1 Definition 36.2 Restricted root systems 36.3 Restriction of a root 37. Symmetric Lie algebras 37.1 Primary subalgebras 37.2 Definition of symmetric Lie algebras 37.3 Natural subalgebras 37.4 Cartan subspaces 37.5 The case of reductive Lie algebras 37.6 Linear forms 38. Semisimple symmetric Lie algebras 38.1 Notations 38.2 Iwasawa decomposition 38.3 Coroots 38.4 Centralizers 38.5 S-triples 38.6 Orbits 38.7 Symmetric invariants 38.8 Double centralizers 38.9 Normalizers 38.10 Distinguished elements 39. Sheets of Lie algebras 39.1 Jordan classes 39.2 Topology of Jordan classes 39.3 Sheets 39.4 Dixmier sheets 39.5 Jordan classes in the symmetric case 39.6 Sheets in the symmetric case 40. Index and linear forms 40.1 Stable linear forms 40.2 Index of a representation 40.3 Some useful inequalities 40.4 Index and semi-direct products 40.5 Heisenberg algebras in semisimple Lie algebras 40.6 Index of Lie subalgebras of Borel subalgebras 40.7 Seaweed Lie algebras 40.8 An upper bound for the index 40.9 Cases where the bound is exact 40.10 On the index of parabolic subalgebras References Index of notations Index of terminologies

Library of Congress subject headings for this publication:

Lie algebras.

Linear algebraic groups.