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| Preface | |
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| Introduction | |
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| The program of Lie | |
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| A result of Galois | |
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| Group theory background | |
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| Approach to solving polynomial equations | |
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| Solution of the quadratic equation | |
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| Solution of the cubic equation | |
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| Solution of the quartic equation | |
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| The quintic cannot be solved | |
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| Example | |
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| Conclusion | |
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| Problems | |
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| Lie groups | |
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| Algebraic properties | |
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| Topological properties | |
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| Unification of algebra and topology | |
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| Unexpected simplification | |
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| Conclusion | |
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| Problems | |
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| Matrix groups | |
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| Preliminaries | |
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| No constraints | |
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| Linear constraints | |
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| Bilinear and quadratic constraints | |
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| Multilinear constraints | |
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| Intersections of groups | |
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| Embedded groups | |
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| Modular groups | |
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| Conclusion | |
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| Problems | |
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| Lie algebras | |
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| Why bother? | |
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| How to linearize a Lie group | |
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| Inversion of the linearization map: EXP | |
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| Properties of a Lie algebra | |
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| Structure constants | |
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| Regular representation | |
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| Structure of a Lie algebra | |
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| Inner product | |
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| Invariant metric and measure on a Lie group | |
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| Conclusion | |
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| Problems | |
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| Matrix algebras | |
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| Preliminaries | |
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| No constraints | |
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| Linear constraints | |
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| Bilinear and quadratic constraints | |
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| Multilinear constraints | |
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| Intersections of groups | |
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| Algebras of embedded groups | |
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| Modular groups | |
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| Basis vectors | |
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| Conclusion | |
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| Problems | |
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| Operator algebras | |
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| Boson operator algebras | |
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| Fermion operator algebras | |
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| First order differential operator algebras | |
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| Conclusion | |
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| Problems | |
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| EXPonentiation | |
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| Preliminaries | |
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| The covering problem | |
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| The isomorphism problem and the covering group | |
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| The parameterization problem and BCH formulas | |
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| EXPonentials and physics | |
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| Conclusion | |
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| Problems | |
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| Structure theory for Lie algebras | |
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| Regular representation | |
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| Some standard forms for the regular representation | |
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| What these forms mean | |
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| How to make this decomposition | |
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| An example | |
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| Conclusion | |
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| Problems | |
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| Structure theory for simple Lie algebras | |
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| Objectives of this program | |
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| Eigenoperator decomposition - secular equation | |
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| Rank | |
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| Invariant operators | |
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| Regular elements | |
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| Semisimple Lie algebras | |
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| Canonical commutation relations | |
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| Conclusion | |
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| Problems | |
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| Root spaces and Dynkin diagrams | |
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| Properties of roots | |
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| Root space diagrams | |
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| Dynkin diagrams | |
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| Conclusion | |
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| Problems | |
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| Real forms | |
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| Preliminaries | |
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| Compact and least compact real forms | |
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| Cartan's procedure for constructing real forms | |
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| Real forms of simple matrix Lie algebras | |
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| Results | |
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| Conclusion | |
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| Problems | |
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| Riemannian symmetric spaces | |
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| Brief review | |
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| Globally symmetric spaces | |
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| Rank | |
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| Riemannian symmetric spaces | |
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| Metric and measure | |
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| Applications and examples | |
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| Pseudo-Riemannian symmetric spaces | |
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| Conclusion | |
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| Problems | |
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| Contraction | |
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| Preliminaries | |
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| Inonu-Wigner contractions | |
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| Simple examples of Inonu-Wigner contractions | |
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| The contraction U(2) to H[subscript 4] | |
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| Conclusion | |
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| Problems | |
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| Hydrogenic atoms | |
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| Introduction | |
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| Two important principles of physics | |
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| The wave equations | |
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| Quantization conditions | |
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| Geometric symmetry SO(3) | |
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| Dynamical symmetry SO(4) | |
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| Relation with dynamics in four dimensions | |
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| DeSitter symmetry SO(4, 1) | |
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| Conformal symmetry SO(4, 2) | |
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| Spin angular momentum | |
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| Spectrum generating group | |
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| Conclusion | |
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| Problems | |
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| Maxwell's equations | |
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| Introduction | |
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| Review of the inhomogeneous Lorentz group | |
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| Subgroups and their representations | |
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| Representations of the Poincare group | |
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| Transformation properties | |
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| Maxwell's equations | |
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| Conclusion | |
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| Problems | |
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| Lie groups and differential equations | |
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| The simplest case | |
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| First order equations | |
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| An example | |
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| Additional insights | |
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| Conclusion | |
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| Problems | |
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| Bibliography | |
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| Index | |