John Stillwell - Naive Lie Theory.pdf - Index of

Index 215
S 2 ,32
not a Lie group, 32
S 3 ,10
as a group, 1, 10, 32
as a matrix group, 32
as special unitary group, 32
homomorphism onto SO(3),23
Hopf fibration of, 26
is not a simple group, 23, 32
is simply connected, 189
S n ,32
scalar product see inner product 13
Schreier, Otto, 73, 115, 150, 201
semisimplicity, 47
of Lie algebras, 138
Sierpinski carpet, 182
simple connectivity, 160, 177
and isomorphism, 186
defined via closed paths, 178
of Lie groups, 186
of R k , 178
of S k , 178
of SU(2), 186, 189
of SU(n) and Sp(n), 190
simplicity
and solvability, 45
Lie’s concept of, 115
of A 5 , 45, 202
of A n , 45, 202
of cross-product algebra, 119
of groups, 31
of Lie algebras, viii, 46, 115
definition, 116
of Lie groups, 48, 115
of sl(n,C), 125
of SO(2m + 1),46
of SO(3), 33, 118, 151
of so(3), 46, 118, 151
of so(n) for n > 4, 130
of sp(n), 133
of su(n), 126
skew field, 21
SL(2,C),92
is noncompact, 92
not the image of exp, 92, 111, 177
universal covering of, 202
SL(n,C), 108, 109
is closed in M n (C), 166
is noncompact, 110
is path-connected, 111
sl(n,C), 109
smoothness, 3, 4, 182
and exponential function, 93, 166
and the tangent space, 183
effected by group structure, 166
of finite groups, 114
of homomorphisms, 183, 191
of manifolds, 3, 114, 182
of matrix groups, 4
of matrix Lie groups, 147
of matrix path, 94
of path, 4, 79, 93, 94
of sequential tangency, 146
SO(2),3
as image of exp, 74
dense subgroup of, 70
is not simply connected, 179, 188
path-connectedness, 53
SO(2m) is not simple, 46, 72
SO(2m + 1) is simple, 46, 70
SO(3),3
and unit quaternions, 33
as Aut(H),44
center of, 61, 151
is not simply connected, 184, 186,
189
is simple, 23, 33, 118, 151
Lie algebra of, 46
same tangents as SU(2), 118, 189
so(3),46
simplicity of, 46, 118, 151
SO(4),23
and quaternions, 23
anomaly of, 47
is not simple, 23, 44, 122