John Stillwell - Naive Lie Theory.pdf - Index of

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128 6 Structure of Lie algebras Our strategy for proving that so(n) is simple is like that used in Section 6.4 to prove that sl(n,C) is simple. It involves two stages: • First we suppose that X is a nonzero member of some ideal I and take Lie brackets of X with suitable basis vectors until we obtain a nonzero multiple of some basis vector in I. • Then, by further Lie bracketing, we show that all basis vectors are in fact in I, soI = so(n). The first stage, as with sl(n,C), selectively nullifies rows and columns until only a nonzero multiple of a basis vector remains. It is a little trickier to do this for so(n), because multiplying by E ij leaves intact two columns (or rows, if one multiplies on the left), rather than one. To nullify all but two, symmetrically positioned, entries we need n > 4, which is no surprise because so(4) is not simple. In the first stage we need to keep track of matrix entries as columns and rows change position, so we introduce a notation that provides number labels to the left of rows and above columns. For example, we write ⎛ i j ⎞ E ij = i ⎜ j ⎝ −1 1 ⎟ ⎠ to indicate that E ij has 1 in the (i, j)-position, −1inthe( j,i)-position, and zeros elsewhere. Now suppose X is the n × n matrix with (i, j)-entry x ij . Multiplying X on the right by E ij and on the left by −E ij ,wefindthat XE ij = ⎛ ⎜ ⎝ i ⎞ −x 1 j x 1i −x 2 j x 2i ⎟ . . ⎠ −x nj x ni j
6.5 Simplicity of so(n) for n > 4 129 and ⎛ ⎞ −E ij X = i −x j1 −x j2 ··· −x jn . j ⎜ x i1 x i2 ··· x in ⎟ ⎝ ⎠ Thus, right multiplication by E ij preserves only column i, which goes to position j, andcolumn j, which goes to position i with its sign changed. Left multiplication by −E ij preserves row i, which goes to position j, and row j, which goes to position i with its sign changed. The Lie bracket of X with E ij is the sum of XE ij and −E ij X, namely [X,E ij ]= ⎛ i j ⎞ −x 1 j x 1i −x 2 j x 2i . . i −x j1 −x j2 ··· −x ji − x ij ··· −x jj + x ii ··· −x jn . . j x i1 x i2 ··· x ii − x jj ··· x ij + x ji ··· x in ⎜ ⎟ ⎝ . . ⎠ −x nj x ni . Note that the (i, j)-and( j,i)-entries are zero when X ∈ so(n) because x ii = x jj = 0 in a skew-symmetric matrix. Likewise, the (i,i)- and( j, j)-entries are zero for a skew-symmetric X, soforX ∈ so(n) we have the simpler formula (*) below. In short, the rule for bracketing a skew-symmetric X with E ij is: • Exchange rows i and j, giving the new row i a minus sign. • Exchange columns i and j, giving the new column i a minus sign. • Put 0 where the new rows and columns meet and 0 everywhere else.
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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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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
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Page 106 and 107: 94 5 The tangent space 5.1 Tangent
Page 108 and 109: 96 5 The tangent space The matrices
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Page 114 and 115: 102 5 The tangent space Exercises A
Page 116 and 117: 104 5 The tangent space To see why
Page 118 and 119: 106 5 The tangent space 5.5 Dimensi
Page 120 and 121: 108 5 The tangent space but not nec
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Page 124 and 125: 112 5 The tangent space However, th
Page 126 and 127: 114 5 The tangent space the set of
Page 128 and 129: 6 Structure of Lie algebras PREVIEW
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Page 152 and 153: 140 7 The matrix logarithm 7.1 Loga
Page 154 and 155: 142 7 The matrix logarithm 7.1.1 Su
Page 156 and 157: 144 7 The matrix logarithm Taking e
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Page 160 and 161: 148 7 The matrix logarithm The log
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Page 164 and 165: 152 7 The matrix logarithm By the t
Page 166 and 167: 154 7 The matrix logarithm The idea
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Page 170 and 171: 158 7 The matrix logarithm Exercise
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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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