Compact Lie Groups

Preface xi 1 Compact Lie Groups 1 1.1 Basic Notions 1 1.1.1 Manifolds 1 1.1.2 Lie Groups 2 1.1.3 Lie Subgroups and Homomorphisms 2 1.1.4 Compact Classical Lie Groups 4 1.1.5 Exercises 6 1.2 Basic Topology 8 1.2.1 Connectedness 8 1.2.2 Simply Connected Cover 10 1.2.3 Exercises 12 1.3 The Double Cover of SO(n) 13 1.3.1 Clifford Algebras 14 1.3.2 Spinn (R) and Pinn (R) 16 1.3.3 Exercises 18 1.4 Integration 19 1.4.1 Volume Forms 19 1.4.2 Invariant Integration 20 1.4.3 Fubini's Theorem 22 1.4.4 Exercises 23 2 Representations 27 2.1 Basic Notions 27 2.1.1 Definitions 27 2.1.2 Examples 28 2.1.3 Exercises 32 2.2 Operations on Representations 34 2.2.1 Constructing New Representations 34 2.2.2 Irreducibility and Schur's Lemma 36 2.2.3 Unitarity 37 2.2.4 Canonical Decomposition 39 2.2.5 Exercises 40 2.3 Examples of Irreducibility 41 2.3.1 SU(2) and Vn(C ) 41 2.3.2 SO(n) and Harmonic Polynomials 42 2.3.3 Spin and Half-Spin Representations 44 2.3.4 Exercises 45 3 Harmonic Analysis 47 3.1 Matrix Coefficients 47 3.1.1 Schur Orthogonality 47 3.1.2 Characters 49 3.1.3 Exercises 52 3.2 Infinite-Dimensional Representations 54 3.2.1 Basic Definitions and Schur's Lemma 54 3.2.2 G-Finite Vectors 56 3.2.3 Canonical Decomposition 58 3.2.4 Exercises 59 3.3 The Peter Weyl Theorem 60 3.3.1 The Left and Right Regular Representation 60 3.3.2 Main Result 64 3.3.3 Applications 66 3.3.4 Exercises 69 3.4 Fourier Theory 70 3.4.1 Convolution 71 3.4.2 Plancherel Theorem 72 3.4.3 Projection Operators and More General Spaces 77 3.4.4 Exercises 79 4 Lie Algebras 81 4.1 Basic Definitions 81 4.1.1 Lie Algebras of Linear Lie Groups 81 4.1.2 Exponential Map 83 4.1.3 Lie Algebras for the Compact Classical Lie Groups 84 4.1.4 Exercises 86 4.2 Further Constructions 88 4.2.1 Lie Algebra Homomorphisms 88 4.2.2 Lie Subgroups and Subalgebras 91 4.2.3 Covering Homomorphisms 92 4.2.4 Exercises 93 5 Abelian Lie Subgroups and Structure 97 5.1 Abelian Subgroups and Subalgebras 97 5.1.1 Maximal Tori and Cartan Subalgebras 97 5.1.2 Examples 98 5.1.3 Conjugacy of Cartan Subalgebras 100 5.1.4 Maximal Torus Theorem 102 5.1.5 Exercises 103 5.2 Structure 105 5.2.1 Exponential Map Revisited 105 5.2.2 Lie Algebra Structure 108 5.2.3 Commutator Theorem 109 5.2.4 Compact Lie Group Structure 110 5.2.5 Exercises 111 6 Roots and Associated Structures 113 6.1 Root Theory 113 6.1.1 Representations of Lie Algebras 113 6.1.2 Complexification of Lie Algebras 115 6.1.3 Weights 116 6.1.4 Roots 117 6.1.5 Compact Classical Lie Group Examples 118 6.1.6 Exercises 120 6.2 The Standard sl(2, C) Triple 123 6.2.1 Cartan Involution 123 6.2.2 Killing Form 124 6.2.3 The Standard sl(2, C) and su(2) Triples 125 6.2.4 Exercises 129 6.3 Lattices 130 6.3.1 Definitions 130 6.3.2 Relations 131 6.3.3 Center and Fundamental Group 132 6.3.4 Exercises 134 6.4 Weyl Group 136 6.4.1 Group Picture 136 6.4.2 Classical Examples 137 6.4.3 Simple Roots and Weyl Chambers 139 6.4.4 The Weyl Group as a Reflection Group 143 6.4.5 Exercises 146 7 Highest Weight Theory 151 7.1 Highest Weights 151 7.1.1 Exercises 154 7.2 Weyl Integration Formula 156 7.2.1 Regular Elements 156 7.2.2 Main Theorem 159 7.2.3 Exercises 162 7.3 Weyl Character Formula 163 7.3.1 Machinery 163 7.3.2 Main Theorem 166 7.3.3 Weyl Denominator Formula 168 7.3.4 Weyl Dimension Formula 168 7.3.5 Highest Weight Classification 169 7.3.6 Fundamental Group 170 7.3.7 Exercises 173 7.4 Borel Weil Theorem 176 7.4.1 Induced Representations 176 7.4.2 Complex Structure on G/T 178 7.4.3 Holomorphic Functions 180 7.4.4 Main Theorem 182 7.4.5 Exercises 184 References 187 Index 193