Matrix Groups An Introduction to Lie Group Theory

Preface......Page 6
3. Tangent Spaces and Lie Algebras......Page 10
Part I. Basic Ideas and Examples......Page 13
1 Real and Complex Matrix Groups......Page 15
1.2 Groups of Matrices as Metric Spaces......Page 17
1.3 Compactness......Page 24
1.4 Matrix Groups......Page 27
1.5 Some Important Examples......Page 30
1.6 Complex Matrices as Real Matrices......Page 41
1.7 Continuous Homomorphisms of Matrix Groups......Page 43
1.8 Matrix Groups for Normed Vector Spaces......Page 45
1.9 Continuous Group Actions......Page 49
2.1 The Matrix Exponentials and Logarithm......Page 57
2.2 Calculating Exponentials and Jordan Form......Page 63
2.3 Differential Equations......Page 67
2.4 One-parameter Subgroups in Matrix Groups......Page 68
2.5 One-parameter Subgroups and Differential Equations......Page 71
3.1 Lie Algebras......Page 79
3.2 Curves, Tangent Spaces and Lie Algebras......Page 83
3.3 The Lie Algebras of Some Matrix Groups......Page 88
3.4 Some Observations on the Exponential Function of a Matrix Group......Page 96
3.5 SO(3) and SU(2)......Page 98
3.6 The Complexification of a Real Lie Algebra......Page 104
8. Homogeneous Spaces......Page 11
4.1 Algebras......Page 111
4.2 Real and Complex Normed Algebras......Page 123
4.3 Linear Algebra over a Division Algebra......Page 125
4.4 The Quaternions......Page 128
4.5 Quaternionic Matrix Groups......Page 132
4.6 Automorphism Groups of Algebras......Page 134
5.1 Real Clifford Algebras......Page 142
5.2 Clifford Groups......Page 151
5.3 Pinor and Spinor Groups......Page 155
5.4 The Centres of Spinor Groups......Page 163
5.5 Finite Subgroups of Spinor Groups......Page 164
6.1 Lorentz Groups......Page 169
6.2 A Principal Axis Theorem for Lorentz Groups......Page 177
6.3 SL_2(C) and the Lorentz Group Lor(3,1)......Page 183
Part II. Matrix Groups as Lie Groups......Page 191
9. Connectivity of Matrix Groups......Page 12
Part III. Compact Connected Lie Groups and their Classification......Page 261
10. Maximal Tori in Compact  Connected Lie Groups......Page 263
11. Semi-simple Factorisation......Page 279
12. Roots Systems, Weyl Groups and Dynkin Diagrams......Page 301
Hints and Solutions to Selected Exercises......Page 315
Bibliography......Page 335
Index......Page 337