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46 2 Groups include “infinitesimal rotations.” Today, we separate out the “infinitesimal rotations” of R 3 in a structure called so(3), theLie algebra of SO(3). The concept of simplicity also makes sense for so(3), and is somewhat easier to establish. Indeed, the infinitesimal elements of any continuous group G form a structure g now called the Lie algebra of G, which captures most of the structure of G but is easier to handle. We discuss “infinitesimal elements,” and their modern counterparts, further in Section 4.3. It was a stroke of luck (or genius) that Lie decided to look at infinitesimal elements, because it enabled him to prove simplicity for whole infinite families of Lie algebras in one fell swoop. (As we will see later, most of the corresponding continuous groups are not quite simple, and one has to tease out certain small subgroups and quotient by them.) Around 1885 Lie proved results so general that they cover all but a finite number of simple Lie algebras—namely, those of the exceptional groups mentioned at the end of Chapter 1 (see Hawkins [2000], pp. 92–98). In the avalanche of Lie’s results, the special case of so(3) and SO(3) seems to have gone unnoticed. It gradually came to light as twentiethcentury books on Lie theory started to work out special cases of geometric interest by way of illustration. In the 1920s, quantum physics also directed attention to SO(3), since rotations in three dimensions are physically significant. Still, it is remarkable that a purely geometric argument for the simplicity of SO(3) took so long to emerge. Perhaps its belated appearance is due to its topological content, namely, the step that depends purely on continuity. The argument hinges on the fact that θ is a continuous function of distance along the great circle PQ, and that such a function takes every value between its extreme values: the so-called intermediate value theorem. The theory of continuity (topology) came after the theory of continuous groups—not surprisingly, since one does not bother to develop a theory of continuity before seeing that it has some content—and applications of topology to group theory were rare before the 1920s. In this book we will present further isolated examples of continuity arguments in Sections 3.2, 3.8, and 7.5 before taking up topology systematically in Chapter 8. Another book with a strongly geometric treatment of SO(3) is Berger [1987]. Volume I of Berger, p. 169, has a simplicity proof for SO(3) similar to the one given here, and it is extended to a simplicity result about SO(n), for n ≥ 5, on p. 170: SO(2m + 1) is simple and the only nontrivial normal subgroup of SO(2m) is {±1}. We arrive at the same result by a different
2.8 Discussion 47 route in Section 7.5. (Our route is longer, but it also takes in the complex and quaternion analogues of SO(n).) Berger treats SO(4) with the help of quaternions on p. 190 of his Volume II, much as we have done here. The quaternion representation of rotations of R 4 was another of Cayley’s discoveries, made in 1855. Lie observed the anomalous structure of SO(4) at the infinitesimal level. He mentions it, in scarcely recognizable form, on p. 683 of Volume III of his 1893 book Theorie der Transformationsgruppen. The anomaly of SO(4) is hidden in some modern treatments of Lie theory, where the concept of simplicity is superseded by the more general concept of semisimplicity. All simple groups are semisimple, and SO(4) is semisimple, so an anomaly is removed by relaxing the concept of “simple” to “semisimple.” However, the concept of semisimplicity makes little sense before one has absorbed the concept of simplicity, and our goal in this book is to understand the simple groups, notwithstanding the anomaly of SO(4).
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Undergraduate Texts in Mathematics
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John Stillwell Naive Lie Theory 123
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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
Page 60 and 61: 3 Generalized rotation groups PREVI
Page 62 and 63: 50 3 Generalized rotation groups Th
Page 64 and 65: 52 3 Generalized rotation groups An
Page 66 and 67: 54 3 Generalized rotation groups th
Page 68 and 69: 56 3 Generalized rotation groups Pa
Page 70 and 71: 58 3 Generalized rotation groups Ho
Page 72 and 73: 60 3 Generalized rotation groups On
Page 74 and 75: 62 3 Generalized rotation groups Ex
Page 76 and 77: 64 3 Generalized rotation groups In
Page 78 and 79: 66 3 Generalized rotation groups Th
Page 80 and 81: 68 3 Generalized rotation groups Ca
Page 82 and 83: 70 3 Generalized rotation groups Pr
Page 84 and 85: 72 3 Generalized rotation groups Ma
Page 86 and 87: 4 The exponential map PREVIEW The g
Page 88 and 89: 76 4 The exponential map course, th
Page 90 and 91: 78 4 The exponential map imaginary
Page 92 and 93: 80 4 The exponential map This const
Page 94 and 95: 82 4 The exponential map 4.4 The Li
Page 96 and 97: 84 4 The exponential map 4.5 The ex
Page 98 and 99: 86 4 The exponential map Definition
Page 100 and 101: 88 4 The exponential map obtained b
Page 102 and 103: 90 4 The exponential map Then subst
Page 104 and 105: 92 4 The exponential map It was dis
Page 106 and 107: 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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