Lie Groups

This textbook provides an essential introduction to Lie groups, presenting the theory from its fundamental principles. Lie groups are a special class of groups that are studied using differential and integral calculus methods. As a mathematical structure, a Lie group combines the algebraic group structure and the differentiable variety structure. Studies of such groups began around 1870 as groups of symmetries of differential equations and the various geometries that had emerged. Since that time, there have been major advances in Lie theory, with ramifications for diverse areas of mathematics and its applications. Each chapter of the book begins with a general, straightforward introduction to the concepts covered; then the formal definitions are presented; and end-of-chapter exercises help to check and reinforce comprehension. Graduate and advanced undergraduate students alike will find in this book a solid yet approachable guide that will help them continue their studies with confidence. Preface 7 Acknowledgement 9 Contents 10 1 Introduction 14 1.1 Exercises 21 Part I Topological Groups 23 Overview 23 2 Topological Groups 25 2.1 Introduction 25 2.2 Neighborhoods of Identity 29 2.3 Metrizable Groups 32 2.4 Homomorphisms 33 2.5 Subgroups 34 2.6 Group Actions 37 2.6.1 Algebraic Description 37 2.6.2 Continuous Actions 40 2.7 Quotient Spaces 41 2.7.1 Quotient Groups 44 2.7.2 Compact and Connected Groups 44 2.8 Homeomorphism G/Gx→G·x 46 2.9 Examples 48 2.10 Exercises 51 3 Haar Measure 54 3.1 Introduction 54 3.2 Construction of Haar Measure 57 3.3 Uniqueness 67 3.4 Modular Function 68 3.5 Exercises 71 4 Representations of Compact Groups 73 4.1 Representations 73 4.2 Schur Orthogonality Relations 78 4.3 Regular Representations 83 4.4 Peter–Weyl Theorem 86 4.5 Exercises 92 Part II Lie Groups and Algebras 93 Overview 93 5 Lie Groups and Lie Algebras 96 5.1 Definition 96 5.2 Lie Algebra of a Lie Group 101 5.2.1 Invariant Vector Fields 102 5.3 Exponential Map 107 5.4 Homomorphisms 111 5.4.1 Representations 114 5.4.2 Adjoint Representations 115 5.5 Ordinary Differential Equations 120 5.6 Haar Measure 121 5.7 Exercises 123 6 Lie Subgroups 126 6.1 Definition and Examples 126 6.2 Lie Subalgebras and Lie Subgroups 129 6.3 Ideals and Normal Subgroups 134 6.4 Limits of Products of Exponentials 136 6.5 Closed Subgroups 138 6.6 Path Connected Subgroups 143 6.7 Manifold Structure on G/H, H Closed 145 6.8 Exercises 149 7 Homomorphisms and Coverings 153 7.1 Homomorphisms 153 7.1.1 Immersions and Submersions 153 7.1.2 Graphs and Differentiability 156 7.2 Extensions of Homomorphisms 157 7.3 Universal Covering 160 7.4 Appendix: Covering Spaces (Overview) 167 7.5 Exercises 168 8 Series Expansions 171 8.1 The Differential of the Exponential Map 171 8.2 The Baker–Campbell–Hausdorff Series 175 8.3 Analytic-Manifold Structure 182 8.4 Exercises 183 Part III Lie Algebras and Simply Connected Groups 185 Overview 185 9 The Affine Group and Semi-Direct Products 188 9.1 Automorphisms of Lie Groups 188 9.2 The Affine Group 195 9.3 Semi-Direct Products 197 9.4 Derived Groups and Lower Central Series 199 9.5 Exercises 203 10 Solvable and Nilpotent Groups 206 10.1 Solvable Groups 206 10.2 Nilpotent Groups 210 10.3 Exercises 215 11 Compact Groups 218 11.1 Compact Lie Algebras 218 11.2 Finite Fundamental Group 223 11.2.1 Extension Theorem 225 11.3 Compact and Complex Lie Algebras 228 11.3.1 Weyl Unitary Trick 228 11.3.2 Dynkin Diagrams 231 11.3.3 Cartan Subalgebras and Regular Elements 233 11.4 Maximal Tori 238 11.5 Center and Roots 242 11.6 Riemannian Geometry 250 11.7 Exercises 252 12 Noncompact Semi-Simple Groups 253 12.1 Cartan Decompositions 253 12.1.1 Cartan Decomposition of a Lie Algebra 254 12.1.2 Global Cartan Decomposition 257 12.2 Iwasawa Decompositions 261 12.2.1 Iwasawa Decomposition of a Lie Algebra 261 12.2.2 Global Iwasawa Decomposition 264 12.3 Classification 267 12.4 Exercises 268 Part IV Transformation Groups 270 Overview 270 13 Lie Group Actions 272 13.1 Group Actions 272 13.1.1 Orbits 276 13.2 Lie–Palais Theorem 279 13.2.1 Families of Vector Fields 285 13.3 Bundles 288 13.3.1 Principal Bundles 288 13.3.2 Associated Bundles 293 13.4 Homogeneous Spaces and Bundles 298 13.5 Exercises 299 14 Invariant Geometry 304 14.1 Complex Manifolds 304 14.1.1 Complex Lie Groups 309 14.2 Differential Forms and de Rham Cohomology 311 14.3 Riemannian Manifolds 322 14.4 Symplectic Manifolds 323 14.4.1 Coadjoint Representation 327 14.4.2 Moment Map 330 14.5 Exercises 340 Part V Appendices 342 A Vector Fields and Lie Brackets 343 A.1 Exercises 348 B Integrability of Distributions 350 B.1 Immersions and Submanifolds 350 B.2 Characteristic Distributions 353 B.3 Maximal Integral Manifolds 359 B.4 Adapted Charts 361 B.5 Integral Manifolds Are Quasi-Regular 363 B.6 Exercises 365 References 366 Index 369