John Stillwell - Naive Lie Theory.pdf - Index of

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208 Index and continuity, 171 and uniform continuity, 172 in general topology, 170 in R k , 169 of O(n) and SO(n), 169 of O(n) and SO(n), 188 of U(n),SU(n), andSp(n), 169 via finite subcover property, 170 complexification, 107, 108 of su(n), 110 of u(n), 109 concatenation, 174, 175, 180 congruence ≡ Lie , 155 conjugation, 14 detects failure to commute, 78 in SU(2), 74 reflected by Lie bracket, 74, 81, 98 rotation by, 14 continuity, 160 and compactness, 171 definition, 166 in general topology, 167 of basic functions, 167 of det, 164, 165 of matrix multiplication, 164 of matrix-valued function, 161 theory of, 46 uniform, 172 and compactness, 172 of deformations, 194 of paths, 194 via open sets, 167 continuous groups, 46 continuous groups see Lie groups 45 convergence, 85 coset, 24 cover, 170 and subcover, 170 covering, 179 and quotient, 201 double, 190 of S 1 by R, 179 of SO(3) by SU(2), 190 universal, 201 cross product, 13 is an operation on R 3 ,13 is Lie bracket on su(2),82 is not associative, 13 satisfies Jacobi identity, 13 curve see path 173 cylinder, 42, 188 Dedekind, Richard, 117 deformation, 160, 177 in small steps, 195 in the plate trick, 184 lifting, 180 with endpoints fixed, 180 Degen, Ferdinand, 22 del Ferro, Scipione, 19 determinant and absolute value, 5 and orientation, 38, 50 and quaternion absolute value, 8 as a homomorphism, 31, 107, 120 is continuous, 164, 165 multiplicative property, 6, 31 of exp, 100, 102 of quaternion matrix, 58, 59 of unitary matrix, 55 dimension, 106 invariance of, 107, 149 of a Lie algebra, 107 of a Lie group, 107 of so(n), 106 of sp(n), 106 of su(n), 106 of u(n), 106 Diophantus, 6 and two-square identity, 18 Arithmetica,6 direct product, 23, 40, 42, 132, 138 direct sum, 132 discreteness, 69 distance and absolute value, 7, 161
Index 209 between matrices, 161 in Euclidean space, 161 distributive law, 7 dot product see inner product 13 Eichler, Martin, viii, 139, 153, 159 eight-square identity, 22 Engel, Friedrich, 92 Euclidean space, 161 Euler, Leonhard, 11 exponential formula, 75 four-square identity, 11, 22 exceptional groups, viii, 22, 45, 46 exp see exponential function 74 exponential function, 74 addition formula, 76, 96, 100, 141 complex, 56, 75 of matrices, 74, 84 definition, 86 provides smooth paths, 93 quaternion, 60, 77 exponential map into Lie groups, 91 into Riemannian manifolds, 92 is not onto SL(2,C), 92, 111, 177 of tangent vectors, 143 onto SO(2), 75 onto SO(3), 99 onto SU(2), 77 finite fields, 202 finite simple groups, 45, 202 finite subcover property, 170 four-square identity, 11 discovered by Euler, 11 Frobenius, Georg, 21, 73 G 2 ,45 Galois theory, 45 Galois, Evariste, 45, 202 Gauss, Carl Friedrich, 11 geodesics, 92 GL(n,C), 108 closed subgroups of, 182 Her All-embracing Majesty, 165 is noncompact, 110 is not simple, 122 is open in M n (C), 166 is path-connected, 111, 175 not closed in M n (C), 165 gl(n,C), 108 is not simple, 122 GL(n,H), 111 gl(n,H), 111 subspaces of, 112 Gleason, Andrew, 159 Graves, John, 22 great circle, 17 reflection in, 17 group abelian, 24, 41, 202 additive notation, 24 affine, 74, 87 center, 61 classical, vii, 80, 82, 93, 113 commutative, 1 continuous, 45 generated by infinitesimals, 91 has finite analogue, 202 coset decomposition of, 25 definition, 24 direct product, 40 discrete, 69, 72, 118, 183 finite, 114, 151 of Lie type, 203 simple, 45, 202 fundamental, 201 general linear, 93, 108 Heisenberg, 72 homomorphism definition, 29 kernel of, 29 preserves structure, 29 identity component, 54 isomorphism, 29 Lie see Lie groups vii
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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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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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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 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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