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Lie Groups, Physics, and Geometry An Introduction for Physicists, Engineers and Chemists

ISBN-10: 0521884004

ISBN-13: 9780521884006

Edition: 2008

Authors: Robert Gilmore

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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.
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Book details

Copyright year: 2008
Publisher: Cambridge University Press
Publication date: 1/17/2008
Binding: Hardcover
Pages: 332
Size: 7.00" wide x 9.75" long x 0.75" tall
Weight: 1.782
Language: English

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