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

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

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

ISBN 0-486-44529-1

Subject Matter


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
Author Index
Subject Index




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
\(\ds \lambda_0 \lambda_i\) \(=\) \(\ds \lambda_i \lambda_0 = \lambda_i\) \(\ds i = 0, 1, 2, 3\)
\(\ds \lambda_i \lambda_i\) \(=\) \(\ds -\lambda_0\)
\(\ds \lambda_1 \lambda_2\) \(=\) \(\ds -\lambda_2 \lambda_1 = \lambda_3\)
\(\ds \lambda_2 \lambda_3\) \(=\) \(\ds -\lambda_3 \lambda_2 = \lambda_1\)
\(\ds \lambda_3 \lambda_1\) \(=\) \(\ds -\lambda_1 \lambda_3 = \lambda_2\)

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