Author	Serre, Jean-Pierre. author
Title	Complex Semisimple Lie Algebras [electronic resource] / by Jean-Pierre Serre
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2001
Connect to	http://dx.doi.org/10.1007/978-3-642-56884-8
Descript	IX, 74 p. online resource

These short notes, already well-known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers, including classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups; it is intended to guide the reader towards further study

I Nilpotent Lie Algebras and Solvable Lie Algebras -- 1. Lower Central Series -- 2. Definition of Nilpotent Lie Algebras -- 3. An Example of a Nilpotent Algebra -- 4. Engelโs Theorems -- 5. Derived Series -- 6. Definition of Solvable Lie Algebras -- 7. Lieโs Theorem -- 8. Cartanโs Criterion -- II Semisimple Lie Algebras (General Theorems) -- 1. Radical and Semisimpiicity -- 2. The Cartan-Killing Criterion -- 3. Decomposition of Semisimple Lie Algebras -- 4. Derivations of Semisimple Lie Algebras -- 5. Semisimple Elements and Nilpotent Elements -- 6. Complete Reducibility Theorem -- 7. Complex Simple Lie Algebras -- 8. The Passage from Real to Complex -- III Cartan Subalgebras -- 1. Definition of Cartan Subalgebras -- 2. Regular Elements: Rank -- 3. The Cartan Subalgebra Associated with a Regular Element -- 4. Conjugacy of Cartan Subalgebras -- 5. The Semisimple Case -- 6. Real Lie Algebras -- IV The Algebra SI2 and Its Representations -- 1. The Lie Algebra sl2 -- 2. Modules, Weights, Primitive Elements -- 3. Structure of the Submodule Generated by a Primitive Element -- 4. The Modules Wm -- 5. Structure of the Finite-Dimensional g-Modules -- 6. Topological Properties of the Group SL2 -- V Root Systems -- 1. Symmetries -- 2. Definition of Root Systems -- 3. First Examples -- 4. The Weyl Group -- 5. Invariant Quadratic Forms -- 6. Inverse Systems -- 7. Relative Position of Two Roots -- 8. Bases -- 9. Some Properties of Bases -- 10. Relations with the Weyl Group -- 11. The Cartan Matrix -- 12. The Coxeter Graph -- 13. Irreducible Root Systems -- 14. Classification of Connected Coxeter Graphs -- 15. Dynkin Diagrams -- 16. Construction of Irreducible Root Systems -- 17. Complex Root Systems -- VI Structure of Semisimple Lie Algebras -- 1. Decomposition of g -- 2. Proof of Theorem 2 -- 3. Borei Subalgebras -- 4. Weyl Bases -- 5. Existence and Uniqueness Theorems -- 6. Chevalleyโs Normalization -- Appendix. Construction of Semisimple Lie Algebras by Generators and Relations -- VII Linear Representations of Semisimple Lie Algebras -- 1. Weights -- 2. Primitive Elements -- 3. Irreducible Modules with a Highest Weight -- 4. Finite-Dimensional Modules -- 5. An Application to the Weyl Group -- 6. Example: sl n+1 -- 7. Characters -- 8. H. Weylโs formula -- VIII Complex Groups and Compact Groups -- 1. Cartan Subgroups -- 2. Characters -- 3. Relations with Representations -- 4. Berel Subgroups -- 5. Construction of Irreducible Representations from Boret Subgroups -- 6. Relations with Algebraic Groups -- 7. Relations with Compact Groups

1. Mathematics
2. Topological groups
3. Lie groups
4. Mathematics
5. Topological Groups
6. Lie Groups