| Preface | p. xi |
| Introduction | p. 1 |
| The program of Lie | p. 1 |
| A result of Galois | p. 2 |
| Group theory background | p. 3 |
| Approach to solving polynomial equations | p. 8 |
| Solution of the quadratic equation | p. 10 |
| Solution of the cubic equation | p. 11 |
| Solution of the quartic equation | p. 15 |
| The quintic cannot be solved | p. 17 |
| Example | p. 18 |
| Conclusion | p. 21 |
| Problems | p. 22 |
| Lie groups | p. 24 |
| Algebraic properties | p. 24 |
| Topological properties | p. 25 |
| Unification of algebra and topology | p. 27 |
| Unexpected simplification | p. 29 |
| Conclusion | p. 29 |
| Problems | p. 30 |
| Matrix groups | p. 34 |
| Preliminaries | p. 34 |
| No constraints | p. 35 |
| Linear constraints | p. 36 |
| Bilinear and quadratic constraints | p. 39 |
| Multilinear constraints | p. 42 |
| Intersections of groups | p. 43 |
| Embedded groups | p. 43 |
| Modular groups | p. 44 |
| Conclusion | p. 46 |
| Problems | p. 47 |
| Lie algebras | p. 55 |
| Why bother? | p. 55 |
| How to linearize a Lie group | p. 56 |
| Inversion of the linearization map: EXP | p. 57 |
| Properties of a Lie algebra | p. 59 |
| Structure constants | p. 61 |
| Regular representation | p. 62 |
| Structure of a Lie algebra | p. 63 |
| Inner product | p. 64 |
| Invariant metric and measure on a Lie group | p. 66 |
| Conclusion | p. 69 |
| Problems | p. 69 |
| Matrix algebras | p. 74 |
| Preliminaries | p. 74 |
| No constraints | p. 74 |
| Linear constraints | p. 75 |
| Bilinear and quadratic constraints | p. 78 |
| Multilinear constraints | p. 80 |
| Intersections of groups | p. 80 |
| Algebras of embedded groups | p. 81 |
| Modular groups | p. 81 |
| Basis vectors | p. 81 |
| Conclusion | p. 83 |
| Problems | p. 83 |
| Operator algebras | p. 88 |
| Boson operator algebras | p. 88 |
| Fermion operator algebras | p. 89 |
| First order differential operator algebras | p. 90 |
| Conclusion | p. 93 |
| Problems | p. 93 |
| EXPonentiation | p. 99 |
| Preliminaries | p. 99 |
| The covering problem | p. 100 |
| The isomorphism problem and the covering group | p. 105 |
| The parameterization problem and BCH formulas | p. 108 |
| EXPonentials and physics | p. 114 |
| Conclusion | p. 119 |
| Problems | p. 120 |
| Structure theory for Lie algebras | p. 129 |
| Regular representation | p. 129 |
| Some standard forms for the regular representation | p. 129 |
| What these forms mean | p. 133 |
| How to make this decomposition | p. 135 |
| An example | p. 136 |
| Conclusion | p. 136 |
| Problems | p. 137 |
| Structure theory for simple Lie algebras | p. 139 |
| Objectives of this program | p. 139 |
| Eigenoperator decomposition - secular equation | p. 140 |
| Rank | p. 143 |
| Invariant operators | p. 143 |
| Regular elements | p. 146 |
| Semisimple Lie algebras | p. 147 |
| Canonical commutation relations | p. 151 |
| Conclusion | p. 153 |
| Problems | p. 154 |
| Root spaces and Dynkin diagrams | p. 159 |
| Properties of roots | p. 159 |
| Root space diagrams | p. 160 |
| Dynkin diagrams | p. 165 |
| Conclusion | p. 168 |
| Problems | p. 168 |
| Real forms | p. 172 |
| Preliminaries | p. 172 |
| Compact and least compact real forms | p. 174 |
| Cartan's procedure for constructing real forms | p. 176 |
| Real forms of simple matrix Lie algebras | p. 177 |
| Results | p. 181 |
| Conclusion | p. 182 |
| Problems | p. 183 |
| Riemannian symmetric spaces | p. 189 |
| Brief review | p. 189 |
| Globally symmetric spaces | p. 190 |
| Rank | p. 191 |
| Riemannian symmetric spaces | p. 192 |
| Metric and measure | p. 193 |
| Applications and examples | p. 194 |
| Pseudo-Riemannian symmetric spaces | p. 197 |
| Conclusion | p. 198 |
| Problems | p. 198 |
| Contraction | p. 205 |
| Preliminaries | p. 205 |
| Inonu-Wigner contractions | p. 206 |
| Simple examples of Inonu-Wigner contractions | p. 206 |
| The contraction U(2) to H[subscript 4] | p. 211 |
| Conclusion | p. 216 |
| Problems | p. 217 |
| Hydrogenic atoms | p. 221 |
| Introduction | p. 221 |
| Two important principles of physics | p. 222 |
| The wave equations | p. 223 |
| Quantization conditions | p. 224 |
| Geometric symmetry SO(3) | p. 227 |
| Dynamical symmetry SO(4) | p. 230 |
| Relation with dynamics in four dimensions | p. 233 |
| DeSitter symmetry SO(4, 1) | p. 235 |
| Conformal symmetry SO(4, 2) | p. 238 |
| Spin angular momentum | p. 243 |
| Spectrum generating group | p. 245 |
| Conclusion | p. 249 |
| Problems | p. 250 |
| Maxwell's equations | p. 259 |
| Introduction | p. 259 |
| Review of the inhomogeneous Lorentz group | p. 261 |
| Subgroups and their representations | p. 262 |
| Representations of the Poincare group | p. 264 |
| Transformation properties | p. 270 |
| Maxwell's equations | p. 273 |
| Conclusion | p. 275 |
| Problems | p. 275 |
| Lie groups and differential equations | p. 284 |
| The simplest case | p. 285 |
| First order equations | p. 286 |
| An example | p. 290 |
| Additional insights | p. 295 |
| Conclusion | p. 302 |
| Problems | p. 303 |
| Bibliography | p. 309 |
| Index | p. 313 |
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