John Stillwell - Naive Lie Theory.pdf - Index of

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Contents 1 Geometry of complex numbers and quaternions 1 1.1 Rotationsoftheplane.................... 2 1.2 Matrix representation of complex numbers ......... 5 1.3 Quaternions......................... 7 1.4 Consequences of multiplicative absolute value . . . .... 11 1.5 Quaternion representation of space rotations ........ 14 1.6 Discussion.......................... 18 2 Groups 23 2.1 Crash course on groups ................... 24 2.2 Crashcourseonhomomorphisms.............. 27 2.3 The groups SU(2) and SO(3) . ............... 32 2.4 Isometries of R n andreflections .............. 36 2.5 Rotations of R 4 andpairsofquaternions.......... 38 2.6 Direct products of groups . . . ............... 40 2.7 The map from SU(2)×SU(2)toSO(4)........... 42 2.8 Discussion.......................... 45 3 Generalized rotation groups 48 3.1 Rotations as orthogonal transformations .......... 49 3.2 The orthogonal and special orthogonal groups . . . .... 51 3.3 The unitary groups ..................... 54 3.4 The symplectic groups ................... 57 3.5 Maximaltoriandcenters .................. 60 3.6 Maximal tori in SO(n), U(n), SU(n), Sp(n) ......... 62 3.7 Centers of SO(n), U(n), SU(n), Sp(n) ............ 67 3.8 Connectednessanddiscreteness .............. 69 3.9 Discussion.......................... 71 xi
xii Contents 4 The exponential map 74 4.1 The exponential map onto SO(2) .............. 75 4.2 The exponential map onto SU(2) .............. 77 4.3 ThetangentspaceofSU(2)................. 79 4.4 The Lie algebra su(2)ofSU(2)............... 82 4.5 The exponential of a square matrix . . . .......... 84 4.6 Theaffinegroupoftheline................. 87 4.7 Discussion.......................... 91 5 The tangent space 93 5.1 Tangent vectors of O(n), U(n), Sp(n) ............ 94 5.2 The tangent space of SO(n) ................. 96 5.3 The tangent space of U(n), SU(n), Sp(n) .......... 99 5.4 Algebraic properties of the tangent space .......... 103 5.5 DimensionofLiealgebras ................. 106 5.6 Complexification ...................... 107 5.7 QuaternionLiealgebras................... 111 5.8 Discussion.......................... 113 6 Structure of Lie algebras 116 6.1 Normal subgroups and ideals . ............... 117 6.2 Idealsandhomomorphisms................. 120 6.3 Classical non-simple Lie algebras . . . .......... 122 6.4 Simplicity of sl(n,C) andsu(n) ............... 124 6.5 Simplicity of so(n) forn > 4 ................ 127 6.6 Simplicity of sp(n) ..................... 133 6.7 Discussion.......................... 137 7 The matrix logarithm 139 7.1 Logarithm and exponential . . ............... 140 7.2 Theexpfunctiononthetangentspace ........... 142 7.3 Limit properties of log and exp ............... 145 7.4 Thelogfunctionintothetangentspace........... 147 7.5 SO(n), SU(n), and Sp(n)revisited ............. 150 7.6 TheCampbell–Baker–Hausdorfftheorem ......... 152 7.7 Eichler’sproofofCampbell–Baker–Hausdorff....... 154 7.8 Discussion.......................... 158
Page 2 and 3: Undergraduate Texts in Mathematics
Page 4 and 5: John Stillwell Naive Lie Theory 123
Page 6 and 7: To Paul Halmos In Memoriam
Page 8 and 9: viii Preface Where my book diverges
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
Page 58 and 59: 46 2 Groups include “infinitesima
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3 Generalized rotation groups PREVI
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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
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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
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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
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154 7 The matrix logarithm The idea
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156 7 The matrix logarithm Next, re
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158 7 The matrix logarithm Exercise
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8 Topology PREVIEW One of the essen
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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
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206 Bibliography Otto Schreier. Abs
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208 Index and continuity, 171 and u
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210 Index Lorentz, 113 matrix, vii,
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212 Index knew SO(4) anomaly, 47 Tr
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214 Index projective space, 185 rea
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216 Index is semisimple, 47 so(4) i
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Undergraduate Texts in Mathematics
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