Normed division algebras: geometrical limitation

[BruceG]

For some time I've been trying to get a geometric appreciation of why normed division algebras only exist in dimensions 1,2,4,8 (namely R,C,H,O).

As always Baez provides the most elegant answer:
http://math.ucr.edu/home/baez/octonions/node6.html"

Allow me to descibe the key point of the proof in case anyone has any better insight to add.

The "division" property of an algebra (ab=0 iff a=0 or b=0) gaurantees that multiplication by unit numbers generates all possible rotations of the unit sphere (e.g. any 2 points on a circle can be reached by rotation by a unit complex number).

From this Baez goes on to show that an n-dimensional normed division algebra must be an irreducible representation of the clifford algebra of dimension n-1.

The result can then be read from the following table:

n -> irreduclible rep of Cliff(n)
0 -> 1
1 -> 2
2 -> 4
3 -> 4
4 -> 8
5 -> 8
6 -> 8
7 -> 8
8 -> 16
9 -> 32
10 -> 64
11 -> 64
12 -> 128
13 -> 128
14 -> 128
15 -> 128

We see that the clifford algebras matrices rapidly become too large to allow the formation of a division algebra.

 

[quasar987]

There is a geometric argument based on, quite surprisingly, the question of paralelizability of the real projective spaces!

It can be shown, using characteristic classes, that the only real projective spaces which are parallelizable are those of dimension 0,1,2,4, and 8.

On the other hand, the existence of a division algebra structure on R^n implies that RP^{n-1} is parallelizable, because n-1 independent sections of the tangent bundle can be explicitly constructed like so:

If B:R^n x R^n -->R^n is a bilinear binary operation on R^n without zero divisors, then for any v in R^n, it is possible to define the map "left multiplication by v" B(v,_) which we may note v*w:=B(v,w). And write v/w for (B(v,_)^-1)(w) [that is, v/w = that "number" such that when you hit it with v from the left gives w.]. Note that for any nonzero v, multiplication by v is a linear automorphism of R^n.
For e1,...,e_n the standard basis of R^n, note that e_i*(x/e_1) are linearly independant for any x. So it suffices to consider the sections of RP^{n-1} defined by
s_i({±x}):=(x-->e_i*(x/e_1) - <e_i*(x/e_1),x>x) for i=2,...,n
where here, the tangent space of RP^{n-1} is seen as the vector bundle \mathrm{Hom}(\tau,\tau^{\perp}) of linear maps between \tau, the tautological line bundle over RP^{n-1}, and its orthogonal complement.

So the only possible dimensions of R^n for which a division algebra structure may exists are those of the form n=2^r. I.e. 1, 2, 4, 8. And we know that they do exists in those dimensions, so that's that.

 

[BruceG]

Thanks for that.

Now I have to work out if this argument is in someway equivalent to Baez' argument or provides an independent restriction. The trouble with a set like {1,2,4,8} is that pure coincidences can occur.