Cover Image
Read Now

Complex Semisimple Lie Algebras

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. Subsequ...

Full description

Bibliographic Details
Main Author: Serre, Jean-Pierre
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 2001, 2001
Edition:1st ed. 2001
Series:Springer Monographs in Mathematics
Subjects:
Topological Groups And Lie Groups
Lie Groups
Topological Groups
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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. Engel’s Theorems
  • 5. Derived Series
  • 6. Definition of Solvable Lie Algebras
  • 7. Lie’s Theorem
  • 8. Cartan’s 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. WeylBases
  • 5. Existence and Uniqueness Theorems
  • 6. Chevalley’s 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. Weyl’s 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

Similar Items