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 spaces
2　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 differentials
3　integral extensions
　3.1　integral dependence
　3.2　integrally closed domains
　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　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 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
　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 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 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 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 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 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 caftan 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 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 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, 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 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 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 jacobson-morosov theorem
　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 richardson's theorem
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 a-root systems
　36.1 definition
　36.2 restricted root systems
　36.3 restriction of a root
37 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 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
　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 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
list of notations
index