John Stillwell - Naive Lie Theory.pdf - Index of

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