Book:Robert Gilmore/Lie Groups, Lie Algebras and Some of their Applications

Subject Matter

 * Lie Groups

Contents

 * Preface


 * 1 Introductory Concepts
 * I. Basic Building Blocks
 * II. Bases
 * III. Mappings, Realizations, Representations


 * 2 The Classical Groups
 * I. General Linear Groups
 * II. Volume Preserving Groups
 * III. Metric Preserving Groups
 * IV. Properties of the Classical Groups


 * 3 Continuous Groups - Lie Groups
 * I. Topological Groups
 * II. An Example
 * III. Additional Necessary Concepts
 * IV. Lie Groups
 * V. The Invariant Integral


 * 4 Lie Groups and Lie Algebras
 * I. Infinitesimal Properties of Lie Groups
 * II. Lie's First Theorem
 * III. Lie's Second Theorem
 * IV. Lie's Third Theorem
 * V. Converses of Lie's Three Theorems
 * VI. Taylor's Theorem for Lie Groups


 * 5 Some Simple Examples
 * I. Relations among some Lie Algebras
 * II. Comparison of Lie Groups
 * III. Representations of $SU(2, c)$
 * IV. Quaternion Covering Group
 * V. Spin and Double-Valuedness - Description of the Electron
 * VI. Noncanonical Parameterizations for $SU(2; c)$


 * 6 Classical Algebras
 * I. Computation of the Algebras
 * II. Topological Properties


 * 7 Lie Algebras and Root Spaces
 * I. General Structure Theory for Lie Algebras
 * II. The Secular Equation
 * III. The Metric
 * IV. Cartan's Criterion
 * V. Canonical Commutation Relations for Semisimple Algebras


 * 8 Root Spaces and Dynkin Diagrams
 * I. Classification of the Simple Root Spaces
 * II. Identification of the Classical Algebras
 * III. Dynkin Diagrams


 * 9 Real Forms
 * I. Algebraic Machinery
 * II. Classification of the Real Forms
 * III. Discussion of Results
 * IV. Properties of Cosets
 * V. Analytical Properties of Cosets
 * VI. Real Forms of the Symmetric Spaces


 * 10 Contractions and Expansions
 * I. Simple Contractions
 * II. Saletan Contractions
 * III. Expansions


 * Bibliogrpahy


 * Author Index


 * Subject Index



Quaternion
Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks