this is a wickedly good book. it's concise (yeah!) and it's well written. it misses out on lots of stuff (spin representations, etc..). but once you read this book you will have the formalism down pat, and then everything else becomes easy.
if you put in the hours to read this book cover to cover -- like sitting down for 3 days straight 8 hours a day, then will learn the stuff. if you don't persevere and get overwhelmed with the stuff that is not clear at the beginning, then you will probably chuck it out the window.
lie groups and lie algebras in 200 pages done in an elegant way that doesn't look like lecture notes cobbled together is pretty impressive.
Author(s): Alexander Kirillov Jr
Series: Cambridge Studies in Advanced Mathematics 113
Publisher: Cambridge University Press
Year: 2008
Language: English
Pages: 236
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Dedication......Page 7
Preface......Page 13
1 Introduction......Page 15
2.1 Reminders from differential geometry......Page 18
2.2 Lie groups, subgroups, and cosets......Page 19
2.4 Action of Lie groups on manifolds and representations......Page 24
2.5 Orbits and homogeneous spaces......Page 26
2.6 Left, right, and adjoint action......Page 28
2.7 Classical groups......Page 30
2.8 Exercises......Page 35
3.1 Exponential map......Page 39
3.2 The commutator......Page 42
3.3 Jacobi identity and the definition of a Lie algebra......Page 44
3.4 Subalgebras, ideals, and center......Page 46
3.5 Lie algebra of vector fields......Page 47
3.6 Stabilizers and the center......Page 50
3.7 Campbell-Hausdorff formula......Page 52
3.8 Fundamental theorems of Lie theory......Page 54
3.9 Complex and real forms......Page 58
3.10.1. Basis and commutation relations......Page 60
3.10.3. Isomorphisms......Page 61
3.11 Exercises......Page 62
4.1 Basic definitions......Page 66
4.2.1. Subrepresentations and quotients......Page 68
4.2.2. Direct sum and tensor product......Page 69
4.2.3. Invariants......Page 70
4.3 Irreducible representations......Page 71
4.4 Intertwining operators and Schur's lemma......Page 73
4.5 Complete reducibility of unitary representations: representations of finite groups......Page 75
4.6 Haar measure on compact Lie groups......Page 76
4.7 Orthogonality of characters and Peter–Weyl theorem......Page 79
4.8 Representations of sl(2,C)......Page 84
4.9 Spherical Laplace operator and the hydrogen atom......Page 89
4.10 Exercises......Page 94
5.1 Universal enveloping algebra......Page 98
5.2 Poincare–Birkhoff–Witt theorem......Page 101
5.3 Ideals and commutant......Page 104
5.4 Solvable and nilpotent Lie algebras......Page 105
5.5 Lie's and Engel's theorems......Page 108
5.6 The radical. Semisimple and reductive algebras......Page 110
5.7 Invariant bilinear forms and semisimplicity of classical Lie algebras......Page 113
5.8 Killing form and Cartan's criterion......Page 115
5.9 Jordan decomposition......Page 118
5.10 Exercises......Page 120
6.1 Properties of semisimple Lie algebras......Page 122
6.2 Relation with compact groups......Page 124
6.3 Complete reducibility of representations......Page 126
6.4 Semisimple elements and toral subalgebras......Page 130
6.5 Cartan subalgebra......Page 133
6.6 Root decomposition and root systems......Page 134
6.7 Regular elements and conjugacy of Cartan subalgebras......Page 140
6.8 Exercises......Page 144
7.1 Abstract root systems......Page 146
7.2 Automorphisms and the Weyl group......Page 148
7.3 Pairs of roots and rank two root systems......Page 149
7.4 Positive roots and simple roots......Page 151
7.5 Weight and root lattices......Page 154
7.6 Weyl chambers......Page 156
7.7 Simple reflections......Page 160
7.8 Dynkin diagrams and classification of root systems......Page 163
7.9 Serre relations and classification of semisimple Lie algebras......Page 168
7.10 Proof of the classification theorem in simply-laced case......Page 171
7.11 Exercises......Page 174
8.1 Weight decomposition and characters......Page 177
8.2 Highest weight representations and Verma modules......Page 181
8.3 Classification of irreducible finite-dimensional representations......Page 185
8.4 Bernstein–Gelfand–Gelfand resolution......Page 188
8.5 Weyl character formula......Page 191
8.6 Multiplicities......Page 196
8.7 Representations of…......Page 197
8.8 Harish–Chandra isomorphism......Page 201
8.9 Proof of Theorem 8.25......Page 206
8.10 Exercises......Page 208
Basic textbooks......Page 211
Infinite-dimensional Lie groups and algebras......Page 212
Unitary infinite-dimensional representations......Page 213
Geometric representation theory......Page 214
Appendix A Root systems and simple Lie algebras......Page 216
Appendix B Sample syllabus......Page 224
Differential geometry......Page 227
Semisimple Lie algebras and root systems......Page 228
Representations of semisimple Lie algebras......Page 229
Bibliography......Page 230
Index......Page 234
