| Preface and Acknowledgments | p. xi |
| Notations | p. xv |
| Lie Groups and Lie Algebras; Introduction | p. 1 |
| Topological Groups | p. 1 |
| Lie Groups | p. 6 |
| Covering Maps and Groups | p. 10 |
| Group Actions and Homogeneous Spaces | p. 15 |
| Lie Algebras | p. 25 |
| Lie Groups | p. 31 |
| Elementary Properties of a Lie Group | p. 31 |
| Taylor's Theorem and the Coefficients of expX expY | p. 39 |
| Correspondence between Lie Subgroups and Subalgebras | p. 45 |
| The Functorial Relationship | p. 48 |
| The Topology of Compact Classical Groups | p. 60 |
| The Iwasawa Decompositions for GL(n, R) and GL(n, C) | p. 67 |
| The Baker-Campbell-Hausdorff Formula | p. 69 |
| Haar Measure and its Applications | p. 89 |
| Haar Measure on a Locally Compact Group | p. 89 |
| Properties of the Modular Function | p. 100 |
| Invariant Measures on Homogeneous Spaces | p. 101 |
| Compact or Finite Volume Quotients | p. 106 |
| Applications | p. 112 |
| Compact Linear Groups and Hilbert's 14th Problem | p. 121 |
| Elements of the Theory of Lie Algebras | p. 127 |
| Basics of Lie Algebras | p. 127 |
| Ideals and Related Concepts | p. 127 |
| Semisimple Lie Algebras | p. 138 |
| Complete Lie Algebras | p. 139 |
| Lie Algebra Representations | p. 140 |
| The Irreducible Representations of sl(2, k) | p. 142 |
| Invariant Forms | p. 145 |
| Complex, Real and Rational Lie Algebras | p. 147 |
| Engel and Lie's Theorems | p. 150 |
| Engel's Theorem | p. 150 |
| Lie's Theorem | p. 153 |
| Cartan's Criterion and Semisimple Lie Algebras | p. 157 |
| Some Algebra | p. 157 |
| Cartan's Solvability Criterion | p. 162 |
| Explicit Computations of Killing Form | p. 166 |
| Further Results on Jordan Decomposition | p. 170 |
| Weyl's Theorem on Complete Reducibility | p. 173 |
| Levi-Malcev Decomposition | p. 179 |
| Reductive Lie Algebras | p. 187 |
| The Jacobson-Morozov Theorem | p. 192 |
| Low Dimensional Lie Algebras over R and C | p. 198 |
| Real Lie Algebras of Compact Type | p. 202 |
| The Structure of Compact Connected Lie Groups | p. 207 |
| Introduction | p. 207 |
| Maximal Tori in Compact Lie Groups | p. 208 |
| Maximal Tori in Compact Connected Lie Groups | p. 210 |
| The Weyl Group | p. 217 |
| What Goes Wrong If G is Not Compact | p. 221 |
| Representations of Compact Lie Groups | p. 223 |
| Introduction | p. 224 |
| The Schur Orthogonality Relations | p. 226 |
| Compact Integral Operators on a Hilbert Space | p. 228 |
| The Peter-Weyl Theorem and its Consequences | p. 234 |
| Characters and Central Functions | p. 243 |
| Induced Representations | p. 250 |
| Some Consequences of Frobenius Reciprocity | p. 255 |
| Symmetric Spaces of Non-compact Type | p. 261 |
| Introduction | p. 261 |
| The Polar Decomposition | p. 264 |
| The Cartan Decomposition | p. 267 |
| The Case of Hyperbolic Space and the Lorentz Group | p. 274 |
| The G-invariant Metric Geometry of P | p. 278 |
| The Conjugacy of Maximal Compact Subgroups | p. 289 |
| The Rank and Two-Point Homogeneous Spaces | p. 294 |
| The Disk Model for Spaces of Rank 1 | p. 299 |
| Exponentiality of Certain Rank 1 Groups | p. 304 |
| Semisimple Lie Algebras and Lie Groups | p. 313 |
| Root and Weight Space Decompositions | p. 313 |
| Cartan Subalgebras | p. 316 |
| Roots of Complex Semisimple Lie Algebras | p. 323 |
| Real Forms of Complex Semisimple Lie Algebras | p. 337 |
| The Iwasawa Decomposition | p. 343 |
| Lattices in Lie Groups | p. 355 |
| Lattices in Euclidean Space | p. 355 |
| GL(n, R)/GL(n, Z) and SL(n, R)/SL(n, Z) | p. 360 |
| Lattices in More General Groups | p. 371 |
| Fundamental Domains | p. 374 |
| Density Results for Cofinite Volume Subgroups | p. 377 |
| Introduction | p. 377 |
| A Density Theorem for Cofinite Volume Subgroups | p. 379 |
| Consequences and Extensions of the Density Theorem | p. 389 |
| Vector Fields | p. 397 |
| The Kronecker Approximation Theorem | p. 403 |
| Properly Discontinuous Actions | p. 407 |
| The Analyticity of Smooth Lie Groups | p. 411 |
| Bibliography | p. 413 |
| Index | p. 421 |
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