Lie Groups Beyond an Introduction

Front Cover
Front Matter
Title Page
Copyright Page
Dedication
Contents
Preface to the Second Edition
Preface to the First Edition
List of Figures
Prerequisites by Chapter
Standard Notation
Introduction: Closed Linear Groups
1. Linear Lie Algebra of a Closed Linear Group
2. Exponential of a Matrix
3. Closed Linear Groups
4. Closed Linear Groups as Lie Groups
5. Homomorphisms
6. Problems
Chapter 1: Lie Algebras and Lie Groups
1. Definitions and Examples
2. Ideals
3. Field Extensions and the Killing Form
4. Semidirect Products of Lie Algebras
5. Solvable Lie Algebras and Lie's Theorem
6. Nilpotent Lie Algebras and Engel's Theorem
7. Cartan's Criterion for Semisimplicity
8. Examples of Semisimple Lie Algebras
9. Representations of sl(2, C)
10. Elementary Theory of Lie Groups
11. Covering Groups
12. Complex Structures
13. Aside on Real-analytic Structures
14. Automorphisms and Derivations
15. Semidirect Products of Lie Groups
16. Nilpotent Lie Groups
17. Classical Semisimple Lie Groups
18. Problems
Chapter 2: Complex Semisimple Lie Algebras
1. Classical Root-space Decompositions
2. Existence of Cartan Subalgebras
3. Uniqueness of Cartan Subalgebras
4. Roots
5. Abstract Root Systems
6. Weyl Group
7. Classification of Abstract Cartan
8. Classification of Nonreduced Abstract
9. Serre Relations
10. Isomorphism Theorem
11. Existence Theorem
12. Problems
Chapter 3: Universal Enveloping Algebra
1. Universal Mapping Property
2. PoincaréBirkhoff—Witt Theorem
3. Associated Graded Algebra
4. Free Lie Algebras
5. Problems
Chapter 4: Compact Lie Groups
1. Examples of Representations
2. Abstract Representation Theory
3. Peter—Weyl Theorem
4. Compact Lie Algebras
5. Centralizers of Tori
6. Analytic Weyl Group
7. Integral Forms
8. Weyl's Theorem
9. Problems
Chapter 5: Finite-Dimensional Representations
1. Weights
2. Theorem of the Highest Weight
3. Verma Modules
4. Complete Reducibility
5. Harish-Chandra Isomorphism
6. Weyl Character Formula
7. Parabolic Subalgebras
8. Application to Compact Lie Groups
9. Problems
Chapter 6: Structure Theory of Semisimple Groups
1. Existence of a Compact Real Form
2. Cartan Decomposition on the Lie Algebra Level
3. Cartan Decomposition on the Lie Group Level
4. Iwasawa Decomposition
5. Uniqueness Properties of the Iwasawa Decomposition
6. Cartan Subalgebras
7. Cayley Transforms
8. Vogan Diagrams
9. Complexification of a Simple Real Lie Algebra
10. Classification of Simple Real Lie Algebras
11. Restricted Roots in the Classification
12. Problems
Chapter 7: Advanced Structure Theory
1. Further Properties of Compact Real Forms
2. Reductive Lie Groups
3. KAK Decomposition
4. Bruhat Decomposition
5. Structure of M
6. Real-rank-one Subgroups
7. Parabolic Subgroups
8. Cartan Subgroups
9. Harish-Chandra Decomposition
10. Problems
Chapter 8: Integration
1. Differential Forms and Measure Zero
2. Haar Measure for Lie Groups
3. Decompositions of Haar Measure
4. Application to Reductive Lie Groups
5. Weyl Integration Formula
6. Problems
Chapter 9: Induced Representations and Branching Theorems
1. Infinite-dimensional Representations of Compact Groups
2. Induced Representations and Frobenius Reciprocity
3. Classical Branching Theorems
4. Overview of Branching
5. Proofs of Classical Branching Theorems
6. Tensor Products and Littlewood—Richardson Coefficients
7. Littlewood's Theorems and an Application
8. Problems
Chapter 10: Prehomogeneous Vector Spaces
1. Definitions and Examples
2. Jacobson—Morozov Theorem
3. Vinberg's Theorem
4. Analysis of Symmetric Tensors
5. Problems
Appendix A: Tensors, Filtrations, and Gradings
1. Tensor Algebra
2. Symmetric Algebra
3. Exterior Algebra
4. Filtrations and Gradings
5. Left Noetherian Rings
Appendix B: Lie’s Third Theorem
1. Levi Decomposition
2. Lie's Third Theorem
3. Ado's Theorem
4. Campbell—Baker—Hausdorff Formula
Appendix C: Data for Simple Lie Algebras
1. Classical Irreducible Reduced Root Systems
2. Exceptional Irreducible Reduced Root Systems
3. Classical Noncompact Simple Real Lie Algebras
4. Exceptional Noncompact Simple Real Lie Algebras
Back Matter
Hints for Solutions of Problems
Historical Notes
References
Index of Notation
Index
Corrections as of June 30, 2023