John Stillwell - Naive Lie Theory.pdf - Index of

1
Geometry of complex
numbers and quaternions
PREVIEW
When the plane is viewed as the plane C of complex numbers, rotation
about O through angle θ is the same as multiplication by the number
e iθ = cos θ + isinθ.
The set of all such numbers is the unit circle or 1-dimensional sphere
S 1 = {z : |z| = 1}.
Thus S 1 is not only a geometric object, but also an algebraic structure;
in this case a group, under the operation of complex number multiplication.
Moreover, the multiplication operation e iθ1 ·e iθ 2
= e i(θ 1+θ 2 ) , and the inverse
operation (e iθ ) −1 = e i(−θ ) , depend smoothly on the parameter θ. This
makes S 1 an example of what we call a Lie group.
However, in some respects S 1 is too special to be a good illustration of
Lie theory. The group S 1 is 1-dimensional and commutative, because multiplication
of complex numbers is commutative. This property of complex
numbers makes the Lie theory of S 1 trivial in many ways.
To obtain a more interesting Lie group, we define the four-dimensional
algebra of quaternions and the three-dimensional sphere S 3 of unit quaternions.
Under quaternion multiplication, S 3 is a noncommutative Lie group
known as SU(2), closely related to the group of space rotations.
J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 1, 1
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