space of all pure imaginary quaternions, which can be identified withR^3 by
w=
w 1
w 2
w 3
∈R^3 ↔~w=w 1 i+w 2 j+w 3 k∈H
Unlike theU(1) case, there’s a non-trivial Lie bracket, the commutator of quater-
nions.
Elements of the groupSp(1) are given by exponentiating such Lie algebra
elements, which we will write in the form
u(θ,w) =eθ~w
whereθ∈Rand~wis a purely imaginary quaternion of unit length. Since
~w^2 = (w 1 i+w 2 j+w 3 k)^2 =−(w^21 +w^22 +w^23 ) =− 1
the exponential can be expanded to show that
eθ~w= cosθ+~wsinθ
Takingθas a parameter, theu(θ,w) give paths inSp(1) going through the
identity atθ= 0, with velocity vector~wsince
d
dθ
u(θ,w)|θ=0= (−sinθ+~wcosθ)|θ=0=~w
We can explicitly evaluate the homomorphism Φ on such elementsu(θ,w)∈
Sp(1), with the result that Φ takesu(θ,w) to a rotation by an angle 2θaround
the axisw:
Theorem 6.1.
Φ(u(θ,w)) =R(2θ,w)
Proof.First consider the special casew=e 3 of rotations about the 3-axis.
u(θ,e 3 ) =eθk= cosθ+ksinθ
and
u(θ,e 3 )−^1 =e−θk= cosθ−ksinθ
so Φ(u(θ,e 3 )) is the rotation that takesv(identified with the quaternion~v=
v 1 i+v 2 j+v 3 k) to
u(θ,e 3 )~vu(θ,e 3 )−^1 =(cosθ+ksinθ)(v 1 i+v 2 j+v 3 k)(cosθ−ksinθ)
=(v 1 (cos^2 θ−sin^2 θ)−v 2 (2 sinθcosθ))i
+ (2v 1 sinθcosθ+v 2 (cos^2 θ−sin^2 θ))j+v 3 k
=(v 1 cos 2θ−v 2 sin 2θ)i+ (v 1 sin 2θ+v 2 cos 2θ)j+v 3 k