Complex Semisimple Lie Algebras / Edition 1

Complex Semisimple Lie Algebras / Edition 1

by Glen A. Jones, Jean-Pierre Serre
     
 

ISBN-10: 3540678271

ISBN-13: 9783540678274

Pub. Date: 01/25/2001

Publisher: Springer Berlin Heidelberg

These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a s ummary of the general properties of nilpotent, solvable, and semisimpl e Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and

…  See more details below

Overview

These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a s ummary of the general properties of nilpotent, solvable, and semisimpl e Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by usin g the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection be tween Lie algebras and Lie groups, and is intended to guide the reader towards further study.

Product Details

ISBN-13:
9783540678274
Publisher:
Springer Berlin Heidelberg
Publication date:
01/25/2001
Series:
Springer Monographs in Mathematics Series
Edition description:
1st ed. 1987. Reprint 2000
Pages:
74
Product dimensions:
9.21(w) x 6.14(h) x 0.25(d)

Related Subjects

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. Weyl Bases.- 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.

Read More

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >