## Lie GroupsThis book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of applying Galois theory to differential equations and in Klein's "Erlanger Programm" of treat ing symmetry groups as the fundamental objects in geometry. Lie's approach to many problems of analysis and geometry was mainly local, that is, valid in local coordinate systems only. At the beginning of the twentieth century E. Cartan and Weyl began a systematic treatment of the global aspects of Lie's theory. Since then this theory has ramified tremendously and now, as the twentieth century is coming to a close, its concepts and methods pervade mathematics and theoretical physics. Despite the plethora of books devoted to Lie groups and Lie algebras we feel there is justification for a text that puts emphasis on Lie's principal idea, namely, geometry treated by a blend of algebra and analysis. Lie groups are geometrical objects whose structure can be described conveniently in terms of group actions and fiber bundles. Therefore our point of view is mainly differential geometrical. We have made no attempt to discuss systematically the theory of infinite-dimensional Lie groups and Lie algebras, which is cur rently an active area of research. We now give a short description of the contents of each chapter. |

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### Contents

Lie Groups and Lie Algebras | 1 |

12 Examples | 6 |

13 The Exponential Map | 16 |

14 The Exponential Map for a Vector Space | 20 |

15 The Tangent Map of Exp | 23 |

16 The Product in Logarithmic Coordinates | 26 |

17 Dynkins Formula | 29 |

18 Lies Fundamental Theorems | 31 |

36 Compact Lie Algebras | 147 |

37 Maximal Tori | 152 |

38 Orbit Structure in the Lie Algebra | 155 |

39 The Fundamental Group | 161 |

310 The Weyl Group as a Reflection Group | 168 |

311 The Stiefel Diagram | 172 |

312 Unitary Groups | 175 |

313 Integration | 179 |

19 The Component of the Identity | 36 |

110 Lie Subgroups and Homomorphisms | 40 |

111 Quotients | 49 |

112 Connected Commutative Lie Groups | 58 |

113 Simply Connected Lie Groups | 62 |

114 Lies Third Fundamental Theorem in Global Form | 72 |

115 Exercises | 81 |

116 Notes | 86 |

References for Chapter One | 90 |

Proper Actions | 93 |

22 Bochners Linearization Theorem | 96 |

23 Slices | 98 |

24 Associated Fiber Bundles | 100 |

25 Smooth Functions on the Orbit Space | 103 |

26 Orbit Types and Local Action Types | 107 |

27 The Stratification by Orbit Types | 111 |

28 Principal and Regular Orbits | 115 |

29 Blowing Up | 122 |

210 Exercises | 126 |

211 Notes | 129 |

References for Chapter Two | 130 |

Compact Lie Groups | 131 |

31 Centralizers | 132 |

32 The Adjoint Action | 139 |

33 Connectedness of Centralizers | 141 |

34 The Group of Rotations and its Covering Group | 143 |

35 Roots and Root Spaces | 144 |

314 The Weyl Integration Theorem | 184 |

315 Nonconnected Groups | 192 |

316 Exercises | 199 |

317 Notes | 202 |

References for Chapter Three | 206 |

Representations of Compact Groups | 209 |

41 Schurs Lemma | 212 |

42 Averaging | 215 |

43 Matrix Coefficients and Characters | 219 |

44 Gtypes | 225 |

45 Finite Groups | 232 |

46 The PeterWeyl Theorem | 233 |

47 Induced Representations | 242 |

48 Reality | 245 |

49 Weyls Character Formula | 252 |

410 Weight Exercises | 263 |

411 Highest Weight Vectors | 285 |

412 The BorelWeil Theorem | 290 |

413 The Nonconnected Case | 306 |

414 Exercises | 318 |

415 Notes | 322 |

326 | |

Appendix | 329 |

Ordinary Differential Equations | 331 |

338 | |

339 | |

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### Common terms and phrases

action of G adjoint algebra g analytic bijective commutative compact Lie group complex-analytic conjugacy classes conjugation connected components connected Lie group Corollary curve defined definition denotes diffeomorphism differential dimension direct sum element equivalent exponential mapping fibration finite follows formula functions g C G G x G G-invariant G/Gx Gc/B GL(V Gprinc Greg hence homomorphism homomorphism of Lie implies induces inner product invariant irreducible representation isomorphism Lemma Let G Lie algebra equal Lie group G Lie subgroup linear mapping linear subspace manifold maximal Abelian subspace maximal torus multiplication nonzero normal subgroup open neighborhood open subset orthogonal polynomial principal orbit type Proof Proposition prove real-analytic representation of G representation TT respectively Section simply connected structure subgroup of G submanifold surjective Theorem topology unique vector fields weights of TT Weyl chamber Weyl group

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