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

Brian C. Hall
Publisher: 
Springer Verlag
Publication Date: 
2003
Number of Pages: 
351
Format: 
Hardcover
Series: 
Graduate Texts in Mathematics
Price: 
59.95
ISBN: 
0-387-40122-9
Category: 
Textbook
[Reviewed by
Gizem Karaali
, on
01/2/2006
]

This book is a great find for those who want to learn about Lie groups or Lie algebras and basics of their representation theory. It is a well-written text which introduces all the basic notions of the theory with many examples and several colored illustrations. The author, Brian Hall of the University of Notre Dame, writes as if he is talking to his best students; without losing rigor and attention to details, he provides many informal explanations, several examples and counterexamples to definitions, discussions and warnings about different conventions, and so on.

But one may ask: Why choose this book among so many others with similar titles? What is so special about this particular one?

The idea is simple: Lie groups and Lie algebras are relevant and useful to many mathematicians (and physicists) with diverse backgrounds. However, at least until recently, these mathematical structures have not been included in the standard curricula of most undergraduate or graduate programs in mathematics. Probably the main reason for this has been the perception that there just are too many prerequisites to even get one's feet wet in the subject and only those few who have been motivated to study the subject in all its details have ventured to test the waters. (How is that for mixed metaphors?) I would go further and dare to say that this is mainly due to the nature of most of the standard texts in the topic, which either choose to focus only on Lie algebras and provide a deep algebraic theory with very little attempt at explaining the geometric connotations (à la Humphreys' Introduction to Lie Algebras and Their Representations), or require a solid background on basic manifold theory and lose a lot of otherwise enthusiastic readers from the beginning (à la Varadarajan's Lie Groups, Lie Algebras, and Their Representations).

However, according to Brian Hall, (and after reading his text, this reviewer  most definitely agrees), there is a way to make almost all the basics of Lie theory and representation theory accessible to a mathematically mature audience who may not have either the prerequisites in manifold theory required to feel comfortable with Varadarajan's text or the interest in a purely algebraic approach like Humphreys'. Hall restricts himself to matrix Lie groups and matrix Lie algebras, which are the main finite dimensional examples. However it is worth noting that he still ends up developing all the theory that one would come across in a more standard text, like the representation theory of semisimple Lie algebras, and in particular the theory of roots and weights.

By restricting to the matrix case, though admittedly losing some generality, the text gains immensely in accessibility. The formal prerequisites reduce now to a solid background in linear algebra (though a detailed appendix covers some of the more advanced topics that are relevant), and a knowledge of some facts about groups and about the convergence of sequences of real or complex numbers. This basically amounts to the usual basic undergraduate courses, and one can even contemplate using the book in a course for an advanced undergraduate audience. However, except for a rather ambitious and highly motivated group of undergraduates, this text could prove to be rather difficult at the undergraduate level. The mathematical maturity required from the reader makes it much more appropriate for beginning graduate students in mathematics or physics. It would also make a great read for mathematicians who want to learn about the subject.


Gizem Karaali teaches at the University of California in Santa Barbara.

Contents
Part I General Theory
1 Matrix Lie Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Definition of a Matrix Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Examples of Matrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The general linear groups GL(n;R) and GL(n;C) . . . . . . 4
1.2.2 The special linear groups SL(n;R) and SL(n;C) . . . . . . . 5
1.2.3 The orthogonal and special orthogonal groups, O(n)
and SO(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 The unitary and special unitary groups, U(n) and SU(n) 6
1.2.5 The complex orthogonal groups, O(n;C) and SO(n;C) . 6
1.2.6 The generalized orthogonal and Lorentz groups . . . . . . . 7
1.2.7 The symplectic groups Sp(n;R), Sp(n;C), and Sp(n) . . . 7
1.2.8 The Heisenberg group H . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.9 The groups R*, C*, S1, R, and Rn . . . . . . . . . . . . . . . . . . . 9
1.2.10 The Euclidean and Poincarïe groups E(n) and P(n;1). . . 9
1.3 Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Examples of compact groups . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Examples of noncompact groups. . . . . . . . . . . . . . . . . . . . . 11
1.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Simple Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.1 Example: SU(2) and SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 The Polar Decomposition for SL(n;R) and SL(n;C) . . . . . . . . . 19
1.8 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Lie Algebras and the Exponential Mapping . . . . . . . . . . . . . . . . 27
2.1 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Computing the Exponential of a Matrix . . . . . . . . . . . . . . . . . . . . 30
X Contents
2.2.1 Case 1: X is diagonalizable . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 Case 2: X is nilpotent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Case 3: X arbitrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 The Matrix Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Further Properties of the Matrix Exponential . . . . . . . . . . . . . . . 35
2.5 The Lie Algebra of a Matrix Lie Group . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Physicists' Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.2 The general linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.3 The special linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.4 The unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.5 The orthogonal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.6 The generalized orthogonal groups . . . . . . . . . . . . . . . . . . 41
2.5.7 The symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.8 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.9 The Euclidean and Poincarïe groups . . . . . . . . . . . . . . . . . . 42
2.6 Properties of the Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 The Exponential Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8.1 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.8.2 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 The Complexification of a Real Lie Algebra . . . . . . . . . . . . . . . . . 56
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 The Baker-Campbell-Hausdorff Formula . . . . . . . . . . . . . . . . . . 63
3.1 The Baker-Campbell-Hausdorff Formula for the Heisenberg
Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 The General Baker-Campbell-Hausdorff Formula . . . . . . . . . . . . 67
3.3 The Derivative of the Exponential Mapping . . . . . . . . . . . . . . . . . 70
3.4 Proof of the Baker-Campbell-Hausdorff Formula . . . . . . . . . . . . 73
3.5 The Series Form of the Baker-Campbell-Hausdorff Formula . . 74
3.6 Group Versus Lie Algebra Homomorphisms . . . . . . . . . . . . . . . . 76
3.7 Covering Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.8 Subgroups and Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Basic Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Why Study Representations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Examples of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 The standard representation . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.2 The trivial representation . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.3 The adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.4 Some representations of SU(2) . . . . . . . . . . . . . . . . . . . . . . 97
4.3.5 Two unitary representations of SO(3) . . . . . . . . . . . . . . . . 99
4.3.6 A unitary representation of the reals . . . . . . . . . . . . . . . . . 100
Contents XI
4.3.7 The unitary representations of the Heisenberg group . . . 100
4.4 The Irreducible Representations of su(2) . . . . . . . . . . . . . . . . . . . . 101
4.5 Direct Sums of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6 Tensor Products of Representations . . . . . . . . . . . . . . . . . . . . . . . . 107
4.7 Dual Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.8 Schur's Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Group Versus Lie Algebra Representations . . . . . . . . . . . . . . . . . . 115
4.10 Complete Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Part II Semisimple Theory
5 The Representations of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2 Weights and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 The Theorem of the Highest Weight . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 An Example: Highest Weight (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6 The Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.7 Weight Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.1 Complete Reducibility and Semisimple Lie Algebras . . . . . . . . . 156
6.2 Examples of Reductive and Semisimple Lie Algebras . . . . . . . . . 161
6.3 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4 Roots and Root Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 Inner Products of Roots and Co-roots . . . . . . . . . . . . . . . . . . . . . . 170
6.6 The Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.8 Positive Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.9 The sl(n;C) Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.9.1 The Cartan subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.9.2 The roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.9.3 Inner products of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.9.4 The Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.9.5 Positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.10 Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Representations of Complex Semisimple Lie Algebras . . . . . . 191
7.1 Integral and Dominant Integral Elements . . . . . . . . . . . . . . . . . . . 192
7.2 The Theorem of the Highest Weight . . . . . . . . . . . . . . . . . . . . . . . 194
7.3 Constructing the Representations I: Verma Modules . . . . . . . . . 200
XII Contents
7.3.1 Verma modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.3.2 Irreducible quotient modules . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3.3 Finite-dimensional quotient modules . . . . . . . . . . . . . . . . . 204
7.3.4 The sl(2;C) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 Constructing the Representations II: The Peter-Weyl Theorem 209
7.4.1 The Peter-Weyl theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.4.2 The Weyl character formula . . . . . . . . . . . . . . . . . . . . . . . . 211
7.4.3 Constructing the representations . . . . . . . . . . . . . . . . . . . . 213
7.4.4 Analytically integral versus algebraically integral
elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.4.5 The SU(2) case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.5 Constructing the Representations III: The Borel-Weil
Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.5.1 The complex-group approach . . . . . . . . . . . . . . . . . . . . . . . 218
7.5.2 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.5.3 The strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.5.4 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.5.5 The SL(2;C) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.6 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.6.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.6.2 The weights and their multiplicities . . . . . . . . . . . . . . . . . . 232
7.6.3 The Weyl character formula and the Weyl dimension
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.6.4 The analytical proof of the Weyl character formula . . . . 236
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8 More on Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.1 Abstract Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.3 Bases and Weyl Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.4 Integral and Dominant Integral Elements . . . . . . . . . . . . . . . . . . . 254
8.5 Examples in Rank Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.5.1 The root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.5.2 Connection with Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 257
8.5.3 The Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.5.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.5.5 Positive roots and dominant integral elements . . . . . . . . . 258
8.5.6 Weight diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
8.6 Examples in Rank Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.7 Additional Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
8.8 The Root Systems of the Classical Lie Algebras . . . . . . . . . . . . . 265
8.8.1 The orthogonal algebras so(2n;C) . . . . . . . . . . . . . . . . . . . 265
8.8.2 The orthogonal algebras so(2n + 1;C) . . . . . . . . . . . . . . . . 266
8.8.3 The symplectic algebras sp(n;C) . . . . . . . . . . . . . . . . . . . . 268
8.9 Dynkin Diagrams and the Classification . . . . . . . . . . . . . . . . . . . . 269
Contents XIII
8.10 The Root Lattice and the Weight Lattice . . . . . . . . . . . . . . . . . . . 273
8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
A A Quick Introduction to Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
A.1 Definition of a Group and Basic Properties . . . . . . . . . . . . . . . . . 279
A.2 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
A.2.1 The trivial group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
A.2.2 The integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
A.2.3 The reals and Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
A.2.4 Nonzero real numbers under multiplication . . . . . . . . . . . 282
A.2.5 Nonzero complex numbers under multiplication . . . . . . . 282
A.2.6 Complex numbers of absolute value 1 under
multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A.2.7 The general linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A.2.8 Permutation group (symmetric group) . . . . . . . . . . . . . . . 283
A.2.9 Integers mod n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A.3 Subgroups, the Center, and Direct Products . . . . . . . . . . . . . . . . 284
A.4 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 285
A.5 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
A.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
B Linear Algebra Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
B.1 Eigenvectors, Eigenvalues, and the Characteristic Polynomial . 291
B.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
B.3 Generalized Eigenvectors and the SN Decomposition . . . . . . . . . 294
B.4 The Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B.5 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B.6 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
B.7 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
B.8 Simultaneous Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
C More on Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.1.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
C.1.3 Differentials of smooth mappings . . . . . . . . . . . . . . . . . . . . 305
C.1.4 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
C.1.5 The flow along a vector field . . . . . . . . . . . . . . . . . . . . . . . . 307
C.1.6 Submanifolds of vector spaces . . . . . . . . . . . . . . . . . . . . . . . 308
C.1.7 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
C.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
C.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
C.2.2 The Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
C.2.3 The exponential mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . 311
C.2.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
XIV Contents
C.2.5 Quotient groups and covering groups . . . . . . . . . . . . . . . . . 312
C.2.6 Matrix Lie groups as Lie groups . . . . . . . . . . . . . . . . . . . . . 313
C.2.7 Complex Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
C.3 Examples of Nonmatrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . 314
C.4 Differential Forms and Haar Measure . . . . . . . . . . . . . . . . . . . . . . 318
D Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
D.1 Tensor Products of sl(2;C) Representations . . . . . . . . . . . . . . . . . 321
D.2 The Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
D.3 More on Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
E Computing Fundamental Groups of Matrix Lie Groups . . . . 331
E.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
E.2 The Universal Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
E.3 Fundamental Groups of Compact Lie Groups I . . . . . . . . . . . . . . 333
E.4 Fundamental Groups of Compact Lie Groups II . . . . . . . . . . . . . 336
E.5 Fundamental Groups of Noncompact Lie Groups . . . . . . . . . . . . 342
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
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