PART I: GENERAL THEORY
1 Matrix Lie Groups.......................................................... 3
1.1 Definitions ............................................................ 3
1.2 Examples ............................................................. 5
1.3 Topological Properties ............................................... 16
1.4 Homomorphisms ..................................................... 21
1.5 Lie Groups............................................................ 25
1.6 Exercises.............................................................. 26
2 The Matrix Exponential ................................................... 31
2.1 The Exponential of a Matrix ......................................... 31
2.2 Computing the Exponential.......................................... 34
2.3 The Matrix Logarithm ............................................... 36
2.4 Further Properties of the Exponential ............................... 40
2.5 The Polar Decomposition ............................................ 42
2.6 Exercises.............................................................. 46
3 Lie Algebras................................................................. 49
3.1 Definitions and First Examples ...................................... 49
3.2 Simple, Solvable, and Nilpotent Lie Algebras...................... 53
3.3 The Lie Algebra of a Matrix Lie Group ............................. 55
3.4 Examples ............................................................. 57
3.5 Lie Group and Lie Algebra Homomorphisms ...................... 60
3.6 The Complexification of a Real Lie Algebra ....................... 65
3.7 The Exponential Map ................................................ 67
3.7 The Exponential Map ................................................ 67
3.8 Consequences of Theorem 3.42 ..................................... 70
3.9 Exercises.............................................................. 73
4 Basic Representation Theory .............................................. 77
4.1 Representations....................................................... 77
4.2 Examples of Representations ........................................ 81
4.3 New Representations from Old ...................................... 84
4.4 Complete Reducibility ............................................... 90
4.5 Schur’s Lemma ....................................................... 94
4.6 Representations of sl.2I C/ .......................................... 96
4.7 Group Versus Lie Algebra Representations......................... 101
4.8 A Nonmatrix Lie Group.............................................. 103
4.9 Exercises.............................................................. 105
The first four chapters of the book cover elementary Lie theory and could be used for an undergraduate course.
5 The Baker–Campbell–Hausdorff Formula and Its Consequences.... 109
5.1 The “Hard” Questions................................................ 109
5.2 An Illustrative Example .............................................. 110
5.3 The Baker–Campbell–Hausdorff Formula .......................... 113
5.4 The Derivative of the Exponential Map ............................. 114
5.5 Proof of the BCH Formula ........................................... 117
5.6 The Series Form of the BCH Formula .............................. 118
5.7 Group Versus Lie Algebra Homomorphisms ....................... 119
5.8 Universal Covers ..................................................... 126
5.9 Subgroups and Subalgebras.......................................... 128
5.10 Lie’s Third Theorem ................................................. 135
5.11 Exercises.............................................................. 135