Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

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Springer Science & Business Media, Aug 7, 2003 - Mathematics - 351 pages
This book provides an introduction to Lie groups, Lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups (not just Lie alge bras) in a way that minimizes the amount of manifold theory needed. Thus, I neither assume a prior course on differentiable manifolds nor provide a con densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time.
 

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Contents

Matrix Lie Groups
5
111 Counterexamples
6
122 The special linear groups SLnℝ and SLn ℂ
7
124 The unitary and special unitary groups Un and SUn
8
126 The generalized orthogonal and Lorentz groups
9
128 The Heisenberg group H
10
129 The groups ℝ ℂ Są ℝ and ℝn
11
13 Compactness
13
Representations of Complex Semisimple Lie Algebras
193
71 Integral and Dominant Integral Elements
194
72 The Theorem of the Highest Weight
196
Verma Modules
202
733 Finitedimensional quotient modules
206
734 The sl2 ℂ case
210
The PeterWeyl Theorem
211
741 The PeterWeyl theorem
212

14 Connectedness
14
15 Simple Connectedness
17
16 Homomorphisms and Isomorphisms
19
SU2 and SO3
20
17 The Polar Decomposition for SLnℝ and SLnℂ
21
18 Lie Groups
22
19 Exercises
25
Lie Algebras and the Exponential Mapping
29
22 Computing the Exponential of a Matrix
32
X is nilpotent
33
X arbitrary
34
24 Further Properties of the Matrix Exponential
37
25 The Lie Algebra of a Matrix Lie Group
40
251 Physicists Convention
41
253 The special linear groups
42
256 The generalized orthogonal groups
43
259 The Euclidean and Poincare groups
44
26 Properties of the Lie Algebra
45
27 The Exponential Mapping
50
28 Lie Algebras
55
281 Structure constants
58
210 Exercises
60
The BakerCampbellHausdorff Formula
65
32 The General BakerCampbellHausdorff Formula
69
33 The Derivative of the Exponential Mapping
72
34 Proof of the BakerCampbellHausdorff Formula
75
35 The Series Form of the BakerCampbellHausdorff Formula
76
36 Group Versus Lie Algebra Homomorphisms
78
37 Covering Groups
82
38 Subgroups and Subalgebras
84
39 Exercises
90
Basic Representation Theory
93
42 Why Study Representations?
96
43 Examples of Representations
97
432 The trivial representation
98
434 Some representations of SU2
99
435 Two unitary representations of SO3
101
436 A unitary representation of the reals
102
44 The Irreducible Representations of su2
103
45 Direct Sums of Representations
108
46 Tensor Products of Representations
109
47 Dual Representations
114
48 Schurs Lemma
115
49 Group Versus Lie Algebra Representations
117
410 Complete Reducibility
120
411 Exercises
123
The Representations of SU3
129
52 Weights and Roots
131
53 The Theorem of the Highest Weight
134
54 Proof of the Theorem
137
Highest Weight 11
142
56 The Weyl Group
144
57 Weight Diagrams
151
Semisimple Lie Algebras
157
61 Complete Reducibility and Semisimple Lie Algebras
158
62 Examples of Reductive and Semisimple Lie Algebras
162
63 Cartan Subalgebras
162
64 Roots and Root Spaces
166
65 Inner Products of Roots and Coroots
172
66 The Weyl Group
175
67 Root Systems
182
68 Positive Roots
183
69 The slnℂ Case
184
693 Inner products of roots
185
694 The Weyl group
186
611 Exercises
187
742 The Weyl character formula
213
743 Constructing the representations
215
744 Analytically integral versus algebraically integral elements
217
745 The SU2 case
218
The BorelWeil Construction
220
752 The setup
222
753 The strategy
224
754 The construction
227
755 The SL2 ℂ case
231
76 Further Results
232
762 The weights and their multiplicities
234
763 The Weyl character formula and the Weyl dimension formula
236
764 The analytical proof of the Weyl character formula
238
More on Roots and Weights
245
82 Duality
250
83 Bases and Weyl Chambers
251
84 Integral and Dominant Integral Elements
256
85 Examples in Rank Two
258
852 Connection with Lie algebras
259
854 Duality
260
856 Weight diagrams
261
86 Examples in Rank Three
264
87 Additional Properties
265
88 The Root Systems of the Classical Lie Algebras
267
882 The orthogonal algebras so2n + 1 ℂ
268
883 The symplectic algebras spn ℂ
270
89 Dynkin Diagrams and the Classification
271
810 The Root Lattice and the Weight Lattice
275
811 Exercises
278
Quick Introduction to Groups
281
A2 Examples of Groups
283
A21 The trivial group
284
A26 Complex numbers of absolute value 1 under multiplication
285
A3 Subgroups the Center and Direct Products
286
A4 Homomorphisms and Isomorphisms
287
A5 Quotient Groups
288
A6 Exercises
291
Linear Algebra Review
293
B2 Diagonalization
295
B3 Generalized Eigenvectors and the SN Decomposition
296
B4 The Jordan Canonical Form
298
B6 Inner Products
299
B7 Dual Spaces
301
More on Lie Groups
305
C12 Tangent space
306
C13 Differentials of smooth mappings
307
C14 Vector fields
308
C15 The flow along a vector field
309
C16 Submanifolds of vector spaces
310
C17 Complex manifolds
311
C22 The Lie algebra
312
C23 The exponential mapping
313
C25 Quotient groups and covering groups
314
C26 Matrix Lie groups as Lie groups
315
C3 Examples of Nonmatrix Lie Groups
316
ClebschGordan Theory for SU2 and the WignerEckart Theorem
323
D2 The WignerEckart Theorem
326
D3 More on Vector Operators
330
Computing Fundamental Groups of Matrix Lie Group
333
E2 The Universal Cover
334
E3 Fundamental Groups of Compact Lie Groups I
335
E4 Fundamental Groups of Compact Lie Groups II
338
E5 Fundamental Groups of Noncompact Lie Groups
344
References
347
Index
349
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About the author (2003)

Brian C. Hall is an Associate Professor of Mathematics at the University of Notre Dame.

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