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Tables of Contents for Lie Groups Beyond an Introduction
Chapter/Section Title
Page #
Page Count
Preface to the Second Edition
xi
Preface to the First Edition
xiii
List of Figures
xvi
Prerequisites by Chapter
xvii
Standard Notation
xviii
Introduction: Closed Linear Groups
1
1
Linear Lie Algebra of a Closed Linear Group
1
5
Exponential of a Matrix
6
3
Closed Linear Groups
9
2
Closed Linear Groups as Lie Groups
11
5
Homomorphisms
16
4
Problems
20
3
Lie Algebras and Lie Groups
23
100
Definitions and Examples
24
5
Ideals
29
4
Field Extensions and the Killing Form
33
5
Semidirect Products of Lie Algebras
38
2
Solvable Lie Algebras and Lie's Theorem
40
5
Nilpotent Lie Algebras and Engel's Theorem
45
4
Cartan's Criterion for Semisimplicity
49
7
Examples of Semisimple Lie Algebras
56
6
Representations of sl(2,C)
62
6
Elementary Theory of Lie Groups
68
13
Covering Groups
81
10
Complex Structures
91
7
Aside on Real-analytic Structures
98
2
Automorphisms and Derivations
100
2
Semidirect Products of Lie Groups
102
4
Nilpotent Lie Groups
106
4
Classical Semisimple Lie Groups
110
8
Problems
118
5
Complex Semisimple Lie Algebras
123
90
Classical Root-space Decompositions
124
5
Existence of Cartan Subalgebras
129
8
Uniqueness of Cartan Subalgebras
137
3
Roots
140
9
Abstract Root Systems
149
13
Weyl Group
162
8
Classification of Abstract Cartan Matrices
170
14
Classification of Nonreduced Abstract Root Systems
184
2
Serre Relations
186
10
Isomorphism Theorem
196
3
Existence Theorem
199
4
Problems
203
10
Universal Enveloping Algebra
213
20
Universal Mapping Property
213
4
Poincare--Birkhoff--Witt Theorem
217
5
Associated Graded Algebra
222
6
Free Lie Algebras
228
1
Problems
229
4
Compact Lie Groups
233
40
Examples of Representations
233
5
Abstract Representation Theory
238
5
Peter--Weyl Theorem
243
5
Compact Lie Algebras
248
3
Centralizers of Tori
251
9
Analytic Weyl Group
260
4
Integral Forms
264
4
Weyl's Theorem
268
1
Problems
269
4
Finite-Dimensional Representations
273
74
Weights
274
5
Theorem of the Highest Weight
279
4
Verma Modules
283
7
Complete Reducibility
290
10
Harish-Chandra Isomorphism
300
14
Weyl Character Formula
314
11
Parabolic Subalgebras
325
8
Application to Compact Lie Groups
333
6
Problems
339
8
Structure Theory of Semisimple Groups
347
86
Existence of a Compact Real Form
348
6
Cartan Decomposition on the Lie Algebra Level
354
7
Cartan Decomposition on the Lie Group Level
361
7
Iwasawa Decomposition
368
10
Uniqueness Properties of the Iwasawa Decomposition
378
6
Cartan Subalgebras
384
5
Cayley Transforms
389
8
Vogan Diagrams
397
9
Complexification of a Simple Real Lie Algebra
406
2
Classification of Simple Real Lie Algebras
408
14
Restricted Roots in the Classification
422
4
Problems
426
7
Advanced Structure Theory
433
90
Further Properties of Compact Real Forms
434
12
Reductive Lie Groups
446
12
K AK Decomposition
458
2
Bruhat Decomposition
460
4
Structure of M
464
6
Real-rank-one Subgroups
470
4
Parabolic Subgroups
474
13
Cartan Subgroups
487
12
Harish-Chandra Decomposition
499
15
Problems
514
9
Integration
523
32
Differential Forms and Measure Zero
523
7
Haar Measure for Lie Groups
530
5
Decompositions of Haar Measure
535
4
Application to Reductive Lie Groups
539
8
Weyl Integration Formula
547
5
Problems
552
3
Induced Representations and Branching Theorems
555
60
Infinite-dimensional Representations of Compact Groups
556
7
Induced Representations and Frobenius Reciprocity
563
5
Classical Branching Theorems
568
3
Overview of Branching
571
6
Proofs of Classical Branching Theorems
577
19
Tensor Products and Littlewood--Richardson Coefficients
596
6
Littlewood's Theorems and an Application
602
7
Problems
609
6
Prehomogeneous Vector Spaces
615
104
Definitions and Examples
616
4
Jacobson--Morozov Theorem
620
6
Vinberg's Theorem
626
6
Analysis of Symmetric Tensors
632
6
Problems
638
1
APPENDICES
A. Tensors, Filtrations, and Gradings
1. Tensor Algebra
639
6
2. Symmetric Algebra
645
6
3. Exterior Algebra
651
3
4. Filtrations and Gradings
654
2
5. Left Noetherian Rings
656
3
B. Lie's Third Theorem
1. Levi Decomposition
659
3
2. Lie's Third Theorem
662
1
3. Ado's Theorem
662
7
4. Campbell-Baker--Hausdorff Formula
669
14
C. Data for Simple Lie Algebras
1. Classical Irreducible Reduced Root: Systems
683
3
2. Exceptional Irreducible Reduced Root Systems
686
7
3. Classical Noncompact Simple Real Lie Algebras
693
13
4. Exceptional Noncompact Simple Real Lie Algebras
706
13
Hints for Solutions of Problems
719
32
Historical Notes
751
32
References
783
16
Index of Notation
799
6
Index
805