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

Robert Gilmore: Lie Groups, Lie Algebras and Some of Their Applications

Published $\text {1974}$, Dover Publications, Inc.

ISBN 0-486-44529-1

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

Next

Errata

Quaternion

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

Every quaternion can be represented in the form

$q = q_0 1 + q_1 \lambda_1 + q_2 \lambda_2 + q_3 \lambda_3$

where the $q_i \, \paren {i = 0, 1, 2, 3}$ are real numbers and the $\lambda_1$ have multiplicative properties defined by

\(\ds \lambda_0 \lambda_i\)

\(=\)

\(\ds \lambda_i \lambda_0 = \lambda_i\)

\(\ds i = 0, 1, 2, 3\)

\(\ds \lambda_i \lambda_i\)

\(=\)

\(\ds -\lambda_0\)

\(\ds \lambda_1 \lambda_2\)

\(=\)

\(\ds -\lambda_2 \lambda_1 = \lambda_3\)

\(\ds \lambda_2 \lambda_3\)

\(=\)

\(\ds -\lambda_3 \lambda_2 = \lambda_1\)

\(\ds \lambda_3 \lambda_1\)

\(=\)

\(\ds -\lambda_1 \lambda_3 = \lambda_2\)

Source work progress

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