李代数和代数群

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 spaces2　Rings and modules　2.1　Ideals　2.2　Prime and maximal ideals　2.3　Rings of fractions and localization　2.4　Localizations 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 differentials3　Integral extensions　3.1　Integral dependence　3.2　Integrally closed domains　3.3　Extensions of prime ideals4　Factorial rings　4.1　Generalities　4.2　Unique factorization　4.3　Principal ideal domains and Euclidean domains　4.4　Polynomials and factorial rings　4.5　Symmetric polynomials　4.6　Resultant and discriminant　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　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 topology7　 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　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　Sheaves of functions　9.1　Sheaves　9.2　Morphisms　9.3　Sheaf associated to a presheaf　9.4　Gluing　 9.5　Ringed space10 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 Examples11 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 sets12 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 variety13 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 variety14 Dimension　14.1 Dimension of varieties　14.2 Dimension and the number of equations .　14.3 System of parameters　14.4 Counterexamples15 Morphisms and dimenion　15.1 Criterion of affineness　15.2 AfIine morphisms　15.3 Finite morphisms　15.4 Factorization and applications　15.5 Dimension of fibres of a morphism　15.6 An example16 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 points17 Normal varieties　17.1 Normal varieties　17.2 Normalization　17.3 Products of normal varieties　17.4 Properties of normal varieties18 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 Relations between two roots　18.6 Examples of root systems　18.7 Base of a root system　18.8 Weyl chambers　18.9 Highest root　18.10 Closed subsets of roots　18.11 Weights　18.12 Graphs　18.13 Dynkin diagrams　18.14 Classification of root systems19 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 Caftan subalgebras20 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 subalgebras21 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 closure22 Ailine 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 one23 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 decomposition24 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 group25 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 groups26 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 groups27 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 Characters28 Borel subgroups, parabolic subgroups, 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 subgroups29 Cartan subalgebras, Borel subalgebras and parabolic　subalgebras　29.1 Generalities　29.2 Cartan subalgebras　29.3 Applications 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 subalgebras30 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 group31 Symmetric invariants　 31.1 Invariants of finite groups　 31.2 Invariant polynomial functions　 31.3 A free module32 S-triples　32.1 Jacobson-Morosov Theorem　32.2 Some lemmas　32.3 Conjugation of S-triples　32.4 Characteristic　32.5 Regular and principal elements33 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 Richardson's Theorem34 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 type35 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 elements36 a-root systems　36.1 Definition　36.2 Restricted root systems　36.3 Restriction of a root37 Symmetric Lie algebras　37.1 Primary subspaces　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 forms38 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 elements39 Sheets of Lie algebras　39.1 Jordan classes　30.2 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 case40 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 subalgebrasReferencesList of notationsIndex

陶威尔编著的《李代数和代数群》内容介绍： The theory of groups and Lie algebras is interesting for many reasons. In the mathematical viewpoint, it employs at the same time algebra, analysis and geometry. On the other hand, it intervenes in other areas of science, in particular in different branches of physics and chemistry. It is an active domain of current research. One of the difficulties that graduate students or mathematicians interested in the theory come across, is the fact that the theory has very much advanced,and consequently, they need to read a vast amount of books and articles before they could tackle interesting problems.