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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… More 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.Less

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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

AuthorTable of Contents

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

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