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 physic8 and chemistry. It is an active domain of current research.
The general theory of algebraic groups is studied in chapter8 21 to 28. The relations between Lie algebras and algebraic groups， which are fundamental to us， are established in chapters 23 and 24. Chapter 29 present8 applications of these relations to tackle the systematic study of Lie algebras. The reader will observe that the geometrical aspects have an important part in this study.In particular， the orbits of points under the action of an algebraic group plays a central role.
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 Ring8 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
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
……
11 Algebraic sets
12 Prevarieties and varieties
13 Projective varieties
14 Dimension
15 Morphisms and dimension
16 Tangent 8paces
17 Normal varieties
18 Root systems
19 Lie algebras
20 Semisimple and reductive Lie algebras
21 Algebraic groups
22 Affine algebraic groups
23 Lie algebra of an algebraic group
24 Correspondence between groups and Lie algebras
25 Homogeneous space8 and quotients
38 Semisimple symmetric Lie algebras
39 Sheets of Lie algebras,
40 Index and linear forms
References
List of notations
Index
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