Division Algebras, Lie Algebras, Lie Groups: 5

Division Algebras, Lie Algebras, Lie Groups and Spinors

5. Division Algebras and More Spheres

The three hypercomplex division algebras are the complex numbers, C (2-dimensional),
quaternions, Q (4-dimensional), and octonions, O (8-dimensional). Without getting too
formal, these algebras are characterized by various properties, all of which boil down to
a simple idea: they are all fundamentally very much like the complex numbers.
And in particular, if A and B are elements of one of these three algebras, and AB = 0,
then either A = 0 or B = 0. Clearly then there can not be any nontrivial projection
operators, for if p (not equal to 1 (or 0, of course)) is a projection operator, then
so is (1-p), and p(1-p) = p - pp = p - p = 0. This contradicts the property above
(no divisors of zero), hence p can not exist.

The other big property has to do with the norm. For any A and B in any of these algebras
we can define norms (real lengths) |A| and |B|, and these satisfy: |AB| = |A||B| (norm of
the product is the product of the norms).

The subsets of norm = 1 of each of these three algebras are unit spheres: respectively,
S¹ (1-sphere = circle); S³ (3-sphere); and S⁷ (7-sphere). Suppose U and V are elements
of one of these spheres (|U|=1; |V|=1), then |UV| = |U||V| = 1. So these spheres are
closed under multiplication - and division (the multiplicative inverse of any A is A^-1 = A*/|A|²).

Well, ok, these three spheres are closed under multiplication, they each possess an
identity (clearly |1| = 1), and there are multiplicative inverses, so they seem to satisfy
everything they'd need to satisfy to be Lie groups. And in fact the 3-sphere (unit
quaternions) is a Lie group (SU(2)), but the 7-sphere (unit octonions) is not, even
though it satisfies all of those nice Lie group properties, and it is parallelizable. The
problem is, octonion multiplication is not associative, and in particular, if U, V and W
are elements of the 7-sphere, then we can NOT in general conclude that (UV)W = U(VW).
And without that it's just not a Lie group (the fact that the position of the parentheses
matters is extremely important, and in what follows we shall exploit this fact to create
from the octonions lots of Lie groups - in particular, SO(8) and SU(3)).

By the way, if U is a unit element of one of these three division algebras, and U is not
the identity, then for any other unit element V, UV ≠ V. That is, multiplication by
U on the k-sphere (k=1,3 or 7) moves every element of the sphere (and smoothly).
Hence the properties defining a division algebra imply these unit k-spheres are
parallelizable. Of the three the 7-sphere is the only surprise: it is the only parallelizable
manifold (with a suitably defined multiplication (I've forgotten the details))
that is not also a Lie group. And while we're at stressing the uniqueness of
things, there are ONLY THREE hypercomplex division algebras (normed, with unity,
and perhaps a condition or two more you'll have to look up), and this is equivalent
to the fact that there are ONLY THREE parallelizable spheres, and this is equivalent
to other unique properties in other mathematical realms. In particular, one can construct
spheres of other dimensions, but none will be parallelizable and none associated with
a division algebra; and one can construct algebraic generalizations of C, Q and O, but
none will yield more division algebras nor be associated with other parallelizable spheres.

We have here a closed context of extremely unique and generative mathematical objects
(generative?: for example, all the classical Lie groups are associated with R, C, Q or O -
R is also a division algebra; its associated unit sphere consists of two points, +1 and -1).
I personally do not believe that there is much to gain by generalizations, for they
do not carry with them the myriad associated special properties. Mathematics seems
to resonate at these special dimensions (1,2,4,8), and no others in the same way or
to the same extent (although we will look at 24 as well, a dimension that resonates
in a different way (for an introduction to that, look at "Sphere Packings" by Conway
and Sloane)).