John Stillwell - Naive Lie Theory.pdf - Index of

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184 8 Topology topology book will show. Their role in Lie theory is also fundamental, and in fact Lie theory provides some of the best illustrations of these concepts. The covering of S 1 by R is one, and we will see more in the next chapter. Closed paths in SO(3) The group SO(3) of rotations of R 3 is a striking example of a matrix Lie group that is not simply connected. We exhibit a closed path in SO(3) that cannot be deformed to a point in an informal demonstration known as the “plate trick.” Imagine carrying a plate of soup in one hand, keeping the plate horizontal to avoid spilling. Now rotate the plate through 360 ◦ , returning it to its original position in space (first three pictures in Figure 8.7). Figure 8.7: The plate trick. The series of positions of the plate up to this stage may be regarded as a continuous path in SO(3). This is because each position is determined by an “axis” in R 3 (the vector from the shoulder to the hand) and an “angle” (the angle through which the plate has turned). This path in SO(3) is closed because the initial and final points are the same (axis, angle) pair. We can “deform” the path by varying the position of the arm and hand between the initial and final positions. But it seems intuitively clear that we cannot
8.8 Discussion 185 deform the path to a single point—because the path creates a full twist in the arm, which cannot be removed by varying the path between the initial and final positions. However, traversing (a deformation of) the path again, asshownin the last three pictures, returns the arm and hand to their initial untwisted state! The topological meaning of this trick is that there is a closed path p in SO(3) that cannot be deformed to a point, whereas p 2 (the result of traversing p twice) can be deformed to a point. This topological property, appropriately called torsion, is actually characteristic of projective spaces, of which SO(3) is one. As we saw in Sections 2.2 and 2.3, SO(3) is the same as the real projective space RP 3 .
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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
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viii Preface Where my book diverges
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Contents 1 Geometry of complex numb
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Contents xiii 8 Topology 160 8.1 Op
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2 1 The geometry of complex numbers
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10 1 The geometry of complex number
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24 2 Groups 2.1 Crash course on gro
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26 2 Groups This algebraic argument
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28 2 Groups is the right coset of H
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30 2 Groups and h ∈ ker ϕ ⇒ ϕ
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32 2 Groups show that the real proj
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34 2 Groups R Q α/2 θ/2 α/2 P Fi
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36 2 Groups 1/2 turn 1/3 turn Figur
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38 2 Groups 2.4.4 Show that reflect
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40 2 Groups 2.5.1 Check that q ↦
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42 2 Groups Exercises If we let x 1
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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
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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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Page 152 and 153: 140 7 The matrix logarithm 7.1 Loga
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Page 198 and 199: 9 Simply connected Lie groups PREVI
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Page 216 and 217: 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
Page 228 and 229: 216 Index is semisimple, 47 so(4) i
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