John Stillwell - Naive Lie Theory.pdf - Index of

214 Index
projective space, 185
real, 32, 33, 185
quantum physics, 46
quaternions, vii, 7
absolute value of, 7
is multiplicative, 9
algebra of, 1, 6
discovered by Hamilton, 10
is skew field, 21
roles in Lie theory, 22
and reflections of R 4 ,38
and rotations, 10, 14, 39
and SO(4),23
automorphisms of, 44
conjugate, 9, 58
inverse, 9
matrix representation, 7
product of, 1, 7
is noncommutative, 8
pure imaginary, 12
as tangent vectors, 79
exponentiation of, 60, 77
spaces of, 22
unit, 10, 14
3-sphere of, 10
and SO(3),33
antipodal, 15
group of, 10
quotient group, 23, 72
definition, 28
homomorphism onto, 28
R 3 as a Lie algebra, 82, 119, 188
as quaternion subspace, 12
rotations of, 10
R 4 ,10
reflections of, 38
rotations of, 23, 36
and quaternions, 39
tiling by 24-cells, 36
R n ,3
isometries of, 18
as products of reflections, 36
rotations of, 3
reflections, 16
and isometries of R n ,36
in great circles, 17
in hyperplanes, 18, 52
linearity of, 38
of R 4 ,38
reverse orientation, 38
representation theory, viii
Riemannian manifolds, 92
rigid motion see isometry 11
Rodrigues, Olinde, 21
root systems, viii, 137
rotations, vii
and quaternions, 10, 14, 15, 35
are isometries, 38
are orientation-preserving, 38
are orthogonal, 3, 49
as product of reflections, 16
form a group, 16
generalized, 59
infinitesimal, 46
of plane, 2
and complex numbers, 3
of R 3 ,10
and quaternions, 15
of R 4 ,23
and quaternions, 39
of R n ,3
definition, 49
of space, 1
and quaternions, 3, 14
do not commute, 9
of tetrahedron, 34
RP 1 ,31
RP 2 , 190
RP 3 , 32, 33, 185
Russell, Bertrand, 193
Ryser, Marc, 37
S 1 ,1
as a group, 1, 32
is not simply connected, 180