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John Stillwell Naive Lie Theory 123
John Stillwell Department of Mathematics University of San Francisco San Francisco, CA 94117 USA stillwell@usfca.edu Editorial Board S. Axler K.A. Ribet Mathematics Department Department of Mathematics San Francisco State University University of California San Francisco, CA 94132 at Berkeley USA Berkeley, CA 94720 axler@sfsu.edu USA ribet@math.berkeley.edu ISBN: 978-0-387-78214-0 e-ISBN: 978-0-387-78215-7 DOI: 10.1007/978-0-387-78214-0 Library of Congress Control Number: 2008927921 Mathematics Subject Classification (2000): 22Exx:22E60 c○ 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 987654321 springer.com
Page 2 and 3: Undergraduate Texts in Mathematics
Page 6 and 7: To Paul Halmos In Memoriam
Page 8 and 9: viii Preface Where my book diverges
Page 10 and 11: Contents 1 Geometry of complex numb
Page 12 and 13: Contents xiii 8 Topology 160 8.1 Op
Page 14 and 15: 2 1 The geometry of complex numbers
Page 22 and 23: 10 1 The geometry of complex number
Page 36 and 37: 24 2 Groups 2.1 Crash course on gro
Page 38 and 39: 26 2 Groups This algebraic argument
Page 40 and 41: 28 2 Groups is the right coset of H
Page 42 and 43: 30 2 Groups and h ∈ ker ϕ ⇒ ϕ
Page 44 and 45: 32 2 Groups show that the real proj
Page 46 and 47: 34 2 Groups R Q α/2 θ/2 α/2 P Fi
Page 48 and 49: 36 2 Groups 1/2 turn 1/3 turn Figur
Page 50 and 51: 38 2 Groups 2.4.4 Show that reflect
Page 52 and 53: 40 2 Groups 2.5.1 Check that q ↦
Page 54 and 55:
42 2 Groups Exercises If we let x 1
Page 56 and 57:
44 2 Groups SO(4) is not simple. Th
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46 2 Groups include “infinitesima
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3 Generalized rotation groups PREVI
Page 62 and 63:
50 3 Generalized rotation groups Th
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52 3 Generalized rotation groups An
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54 3 Generalized rotation groups th
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56 3 Generalized rotation groups Pa
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58 3 Generalized rotation groups Ho
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60 3 Generalized rotation groups On
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62 3 Generalized rotation groups Ex
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64 3 Generalized rotation groups In
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66 3 Generalized rotation groups Th
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68 3 Generalized rotation groups Ca
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70 3 Generalized rotation groups Pr
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72 3 Generalized rotation groups Ma
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4 The exponential map PREVIEW The g
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76 4 The exponential map course, th
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78 4 The exponential map imaginary
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80 4 The exponential map This const
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82 4 The exponential map 4.4 The Li
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84 4 The exponential map 4.5 The ex
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86 4 The exponential map Definition
Page 100 and 101:
88 4 The exponential map obtained b
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90 4 The exponential map Then subst
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92 4 The exponential map It was dis
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94 5 The tangent space 5.1 Tangent
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96 5 The tangent space The matrices
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98 5 The tangent space as in ordina
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100 5 The tangent space Conversely,
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102 5 The tangent space Exercises A
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104 5 The tangent space To see why
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106 5 The tangent space 5.5 Dimensi
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108 5 The tangent space but not nec
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110 5 The tangent space Conversely,
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112 5 The tangent space However, th
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114 5 The tangent space the set of
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6 Structure of Lie algebras PREVIEW
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118 6 Structure of Lie algebras isa
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120 6 Structure of Lie algebras An
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122 6 Structure of Lie algebras 6.3
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124 6 Structure of Lie algebras We
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126 6 Structure of Lie algebras whi
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128 6 Structure of Lie algebras Our
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130 6 Structure of Lie algebras [X,
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132 6 Structure of Lie algebras l
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134 6 Structure of Lie algebras and
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136 6 Structure of Lie algebras If
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138 6 Structure of Lie algebras of
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140 7 The matrix logarithm 7.1 Loga
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142 7 The matrix logarithm 7.1.1 Su
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144 7 The matrix logarithm Taking e
Page 158 and 159:
146 7 The matrix logarithm If A(t)
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148 7 The matrix logarithm The log
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150 7 The matrix logarithm 7.4.1 Sh
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152 7 The matrix logarithm By the t
Page 166 and 167:
154 7 The matrix logarithm The idea
Page 168 and 169:
156 7 The matrix logarithm Next, re
Page 170 and 171:
158 7 The matrix logarithm Exercise
Page 172 and 173:
8 Topology PREVIEW One of the essen
Page 174 and 175:
162 8 Topology The set N ε (P) is
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164 8 Topology 8.2 Closed matrix gr
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166 8 Topology Matrix Lie groups Wi
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168 8 Topology also a continuous fu
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170 8 Topology Pick, say, the leftm
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172 8 Topology true that f −1 (
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174 8 Topology describing specific
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176 8 Topology These roots represen
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178 8 Topology The restriction of d
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180 8 Topology can divide [0,1] int
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182 8 Topology 8.8 Discussion Close
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184 8 Topology topology book will s
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9 Simply connected Lie groups PREVI
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188 9 Simply connected Lie groups T
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190 9 Simply connected Lie groups I
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192 9 Simply connected Lie groups f
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194 9 Simply connected Lie groups 9
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196 9 Simply connected Lie groups T
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198 9 Simply connected Lie groups W
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200 9 Simply connected Lie groups L
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202 9 Simply connected Lie groups L
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Bibliography J. Frank Adams. Lectur
Page 218 and 219:
206 Bibliography Otto Schreier. Abs
Page 220 and 221:
208 Index and continuity, 171 and u
Page 222 and 223:
210 Index Lorentz, 113 matrix, vii,
Page 224 and 225:
212 Index knew SO(4) anomaly, 47 Tr
Page 226 and 227:
214 Index projective space, 185 rea
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216 Index is semisimple, 47 so(4) i
Page 230:
Undergraduate Texts in Mathematics
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