John Stillwell - Naive Lie Theory.pdf - Index of

42 2 Groups
Exercises
If we let x 1 ,x 2 ,x 3 ,x 4 be the coordinates along mutually orthogonal axes in R 4 ,
then it is possible to “rotate” the x 1 and x 2 axes while keeping the x 3 and x 4 axes
fixed.
2.6.1 Writea4× 4 matrix for the transformation that rotates the (x 1 ,x 2 )-plane
through angle θ while keeping the x 3 -andx 4 -axes fixed.
2.6.2 Writea4× 4 matrix for the transformation that rotates the (x 3 ,x 4 )-plane
through angle φ while keeping the x 1 -andx 2 -axes fixed.
2.6.3 Observe that the rotations in Exercise 2.6.1 form an S 1 , as do the rotations
in Exercise 2.6.2, and deduce that SO(4) contains a subgroup isomorphic
to T 2 .
The groups of the form R m × T n may be called “generalized cylinders,” based
on the simplest example R × S 1 .
2.6.4 Why is it appropriate to call the group R × S 1 a cylinder?
The notation S n is unfortunately not compatible with the direct product notation
(at least not the way the notation R n is).
2.6.5 Explain why S 3 = SU(2) is not the same group as S 1 × S 1 × S 1 .
2.7 The map from SU(2)×SU(2) to SO(4)
In Section 2.5 we showed that the rotations of R 4 are precisely the maps
q ↦→ vqw, wherev and w run through all the unit quaternions. Since v −1
is a unit quaternion if and only if v is, it is equally valid to represent each
rotation of R 4 by a map of the form q ↦→ v −1 qw, wherev and w are unit
quaternions. The latter representation is more convenient for what comes
next.
The pairs of unit quaternions (v,w) form a group under the operation
defined by
(v 1 ,w 1 ) · (v 2 ,w 2 )=(v 1 v 2 ,w 1 w 2 ),
where the products v 1 v 2 and w 1 w 2 on the right side are ordinary quaternion
products. Since the v come from the group SU(2) of unit quaternions, and
the w likewise, the group of pairs (v,w) is the direct product SU(2)×SU(2)
of SU(2) with itself.
The map that sends each pair (v,w) ∈ SU(2) × SU(2) to the rotation
q ↦→ v −1 qw in SO(4) is a homomorphism ϕ :SU(2) × SU(2) → SO(4).
This is because