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Lie Groups, Physics, and Geometry

Book Description

Describing many of the most important aspects of Lie group theory, this book presents the subject in a 'hands on' way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Introduction
    1. 1.1 The program of Lie
    2. 1.2 A result of Galois
    3. 1.3 Group theory background
    4. 1.4 Approach to solving polynomial equations
    5. 1.5 Solution of the quadratic equation
    6. 1.6 Solution of the cubic equation
    7. 1.7 Solution of the quartic equation
    8. 1.8 The quintic cannot be solved
    9. 1.9 Example
    10. 1.10 Conclusion
    11. 1.11 Problems
  7. 2 Lie groups
    1. 2.1 Algebraic properties
    2. 2.2 Topological properties
    3. 2.3 Unification of algebra and topology
    4. 2.4 Unexpected simplification
    5. 2.5 Conclusion
    6. 2.6 Problems
  8. 3 Matrix groups
    1. 3.1 Preliminaries
    2. 3.2 No constraints
    3. 3.3 Linear constraints
    4. 3.4 Bilinear and quadratic constraints
    5. 3.5 Multilinear constraints
    6. 3.6 Intersections of groups
    7. 3.7 Embedded groups
    8. 3.8 Modular groups
    9. 3.9 Conclusion
    10. 3.10 Problems
  9. 4 Lie algebras
    1. 4.1 Why bother?
    2. 4.2 How to linearize a Lie group
    3. 4.3 Inversion of the linearization map: EXP
    4. 4.4 Properties of a Lie algebra
    5. 4.5 Structure constants
    6. 4.6 Regular representation
    7. 4.7 Structure of a Lie algebra
    8. 4.8 Inner product
    9. 4.9 Invariant metric and measure on a Lie group
    10. 4.10 Conclusion
    11. 4.11 Problems
  10. 5 Matrix algebras
    1. 5.1 Preliminaries
    2. 5.2 No constraints
    3. 5.3 Linear constraints
    4. 5.4 Bilinear and quadratic constraints
    5. 5.5 Multilinear constraints
    6. 5.6 Intersections of groups
    7. 5.7 Algebras of embedded groups
    8. 5.8 Modular groups
    9. 5.9 Basis vectors
    10. 5.10 Conclusion
    11. 5.11 Problems
  11. 6 Operator algebras
    1. 6.1 Boson operator algebras
    2. 6.2 Fermion operator algebras
    3. 6.3 First order differential operator algebras
    4. 6.4 Conclusion
    5. 6.5 Problems
  12. 7 EXPonentiation
    1. 7.1 Preliminaries
    2. 7.2 The covering problem
    3. 7.3 The isomorphism problem and the covering group
    4. 7.4 The parameterization problem and BCH formulas
    5. 7.5 EXPonentials and physics
    6. 7.6 Conclusion
    7. 7.7 Problems
  13. 8 Structure theory for Lie algebras
    1. 8.1 Regular representation
    2. 8.2 Some standard forms for the regular representation
    3. 8.3 What these forms mean
    4. 8.4 How to make this decomposition
    5. 8.5 An example
    6. 8.6 Conclusion
    7. 8.7 Problems
  14. 9 Structure theory for simple Lie algebras
    1. 9.1 Objectives of this program
    2. 9.2 Eigenoperator decomposition – secular equation
    3. 9.3 Rank
    4. 9.4 Invariant operators
    5. 9.5 Regular elements
    6. 9.6 Semisimple Lie algebras
    7. 9.7 Canonical commutation relations
    8. 9.8 Conclusion
    9. 9.9 Problems
  15. 10 Root spaces and Dynkin diagrams
    1. 10.1 Properties of roots
    2. 10.2 Root space diagrams
    3. 10.3 Dynkin diagrams
    4. 10.4 Conclusion
    5. 10.5 Problems
  16. 11 Real forms
    1. 11.1 Preliminaries
    2. 11.2 Compact and least compact real forms
    3. 11.3 Cartan’s procedure for constructing real forms
    4. 11.4 Real forms of simple matrix Lie algebras
    5. 11.5 Results
    6. 11.6 Conclusion
    7. 11.7 Problems
  17. 12 Riemannian symmetric spaces
    1. 12.1 Brief review
    2. 12.2 Globally symmetric spaces
    3. 12.3 Rank
    4. 12.4 Riemannian symmetric spaces
    5. 12.5 Metric and measure
    6. 12.6 Applications and examples
    7. 12.7 Pseudo-Riemannian symmetric spaces
    8. 12.8 Conclusion
    9. 12.9 Problems
  18. 13 Contraction
    1. 13.1 Preliminaries
    2. 13.2 Inönü–Wigner contractions
    3. 13.3 Simple examples of Inönü–Wigner contractions
    4. 13.4 The contraction U (2) → H4
    5. 13.5 Conclusion
    6. 13.6 Problems
  19. 14 Hydrogenic atoms
    1. 14.1 Introduction
    2. 14.2 Two important principles of physics
    3. 14.3 The wave equations
    4. 14.4 Quantization conditions
    5. 14.5 Geometric symmetry SO(3)
    6. 14.6 Dynamical symmetry SO(4)
    7. 14.7 Relation with dynamics in four dimensions
    8. 14.8 DeSitter symmetry SO(4, 1)
    9. 14.9 Conformal symmetry SO(4, 2)
    10. 14.10 Spin angular momentum
    11. 14.11 Spectrum generating group
    12. 14.12 Conclusion
    13. 14.13 Problems
  20. 15 Maxwell’s equations
    1. 15.1 Introduction
    2. 15.2 Review of the inhomogeneous Lorentz group
    3. 15.3 Subgroups and their representations
    4. 15.4 Representations of the Poincaré group
    5. 15.5 Transformation properties
    6. 15.6 Maxwell’s equations
    7. 15.7 Conclusion
    8. 15.8 Problems
  21. 16 Lie groups and differential equations
    1. 16.1 The simplest case
    2. 16.2 First order equations
    3. 16.3 An example
    4. 16.4 Additional insights
    5. 16.5 Conclusion
    6. 16.6 Problems
  22. Bibliography
  23. Index