Author | Serre, Jean-Pierre. author |
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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 |
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