Naive Lie Theory

by Stillwell, John

• ISBN13: 9780387782140
• ISBN10: 0387782141
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-07-04
• Publisher: Springer Verlag

Summary

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called 'classical groups' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.

Author Biography

John Stillwell is Professor of Mathematics at the University of San Francisco

Table of Contents

Geometry of complex numbers and quaternions	p. 1
Rotations of the plane	p. 2
Matrix representation of complex numbers	p. 5
Quaternions	p. 7
Consequences of multiplicative absolute value	p. 11
Quaternion representation of space rotations	p. 14
Discussion	p. 18
Groups	p. 23
Crash course on groups	p. 24
Crash course on homomorphisms	p. 27
The groups SU(2) and SO(3)	p. 32
Isometries of R[superscript n] and reflections	p. 36
Rotations of R[superscript 4] and pairs of quaternions	p. 38
Direct products of groups	p. 40
The map from SU(2)xSU(2) to SO(4)	p. 42
Discussion	p. 45
Generalized rotation groups	p. 48
Rotations as orthogonal transformations	p. 49
The orthogonal and special orthogonal groups	p. 51
The unitary groups	p. 54
The symplectic groups	p. 57
Maximal tori and centers	p. 60
Maximal tori in SO(n), U(n), SU(n), Sp(n)	p. 62
Centers of SO(n), U(n), SU(n), Sp(n)	p. 67
Connectedness and discreteness	p. 69
Discussion	p. 71
The exponential map	p. 74
The exponential map onto SO(2)	p. 75
The exponential map onto SU(2)	p. 77
The tangent space of SU(2)	p. 79
The Lie algebra su(2) of SU(2)	p. 82
The exponential of a square matrix	p. 84
The affine group of the line	p. 87
Discussion	p. 91
The tangent space	p. 93
Tangent vectors of O(n), U(n), Sp(n)	p. 94
The tangent space of SO(n)	p. 96
The tangent space of U(n), SU(n), Sp(n)	p. 99
Algebraic properties of the tangent space	p. 103
Dimension of Lie algebras	p. 106
Complexification	p. 107
Quaternion Lie algebras	p. 111
Discussion	p. 113
Structure of Lie algebras	p. 116
Normal subgroups and ideals	p. 117
Ideals and homomorphisms	p. 120
Classical non-simple Lie algebras	p. 122
Simplicity of sl(n, C) and su(n)	p. 124
Simplicity of so(n) for n > 4	p. 127
Simplicity of sp(n)	p. 133
Discussion	p. 137
The matrix logarithm	p. 139
Logarithm and exponential	p. 140
The exp function on the tangent space	p. 142
Limit properties of log and exp	p. 145
The log function into the tangent space	p. 147
SO(n), SU(n), and Sp(n) revisited	p. 150
The Campbell-Baker-Hausdorff theorem	p. 152
Eichler's proof of Campbell-Baker-Hausdorff	p. 154
Discussion	p. 158
Topology	p. 160
Open and closed sets in Euclidean space	p. 161
Closed matrix groups	p. 164
Continuous functions	p. 166
Compact sets	p. 169
Continuous functions and compactness	p. 171
Paths and path-connectedness	p. 173
Simple connectedness	p. 177
Discussion	p. 182
Simply connected Lie groups	p. 186
Three groups with tangent space R	p. 187
Three groups with the cross-product Lie algebra	p. 188
Lie homomorphisms	p. 191
Uniform continuity of paths and deformations	p. 194
Deforming a path in a sequence of small steps	p. 195
Lifting a Lie algebra homomorphism	p. 197
Discussion	p. 201
Bibliography	p. 204
Index	p. 207
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