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

The set of all
unit complexes (norm 1 and can be written in the form exp(iθ))
is closed

under multiplication (exp(ia)exp(ib) = exp(i(a+b)). This set fulfills all the
requirements

of a Lie group (roughly: multiplicative closure; inverses; identity; and the
full set is a

nice very smooth continuous set (manifold); We needn't go into elaborate rigorous

detail - we'll come to know them when we see them). This Lie group is U(1).
The

Lie algebra associated with U(1) is the set from which U(1) is derived via exponentiation:

hence the Lie algebra of U(1), which we denote u(1), is the set {iθ: θ
real} - the set

of purely imaginary complex numbers.

The associated
manifold is the circle, the 1-dimensional sphere. This manifold

has a special property: it's parallelizable (all Lie groups are, but only three
spheres -

although two of these three parallelizable spheres are also Lie groups - more
anon).

This can be visualized as follows: a manifold is parallelizable if it is possible
to set

all its points in smooth flowing motion at the same time (a good example of
a

nonparallelizable manifold is the 2-sphere (surface of a ball). There is always
at least

one point on the surface of a ball that is stationary: it's impossible to set
them all in

smooth motion at once (the fact that the earth has poles is related to this)).
It must

be further specified that at any point of the manifold the smooth flowing motion
can

flow in any direction. Clearly this is an area where mathematical rigor would
be of some

use, but our intention here is simply to introduce the concept as a way of highlighting

the exceptional nature of the 1-, 3- and 7-spheres.

Spin a 1-sphere
and all the points move at once (not so a 2-sphere). It's clearly

parallelizable.

There are two ways
to view this parallelizablility: the 1-sphere is also the Lie group U(1),

and all Lie groups are parallelizable; and the 1-sphere is the set of all unit
complex

numbers, and the complex numbers are a division algebra - and it can be shown
that the set

of all unit elements of any such division algebra must be parallelizable. There
are

three hypercomplex division algebras; and there are three nontrivial parallelizable

unit spheres - three spheres on which all points may move at once.

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