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**Division Algebras,
Lie Algebras, Lie Groups and Spinors**

One of the most
beautiful equations in all of mathematics is the exponential

of a pure imaginary complex number:

A really big point to make here is the following: the only property of the complex

unit i used by this equation is i^{2}= -1. If u is an element of an algebra that has an

identity (1), and u^{2}= -1, then exp(uθ) = cos(θ) + u sin(θ). We shall use this fact

repeatedly in what follows.

The norm squared
(length squared) of the complex number z = x + iy is:

zz* = (x + iy)(x - iy) = x^{2} + y^{2}, where z* = (x - iy)
is the complex conjugate of z.

The complex conjugate of exp(iθ) is exp(-iθ)
= cos (θ) - i sin(θ),
so the norm

squared of exp(iθ) is

That is, the set
of all exp(iθ) is just the unit sphere (circle) in
the 2-dimensional

complex plane. This set plays many roles. In particular, it is the 1-sphere,
one of

three nontrivial parallelizable spheres (more anon); and it is the Lie group
U(1).

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