# Normed division algebras: geometrical limitation

• BruceG
In summary: Math is all about making the right deductions and I'm not sure if this is the right deduction. In summary, the proof shows that normed division algebras only exist in dimensions 1,2,4,8 (namely R,C,H,O). They cannot exist in higher dimensions for reasons that are based on the question of paralelizability of the real projective spaces.
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.

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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.

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.

## 1. What is a normed division algebra?

A normed division algebra is a mathematical structure that combines the properties of a normed vector space and a division algebra. This means that it is a vector space where multiplication and division are defined, and there is a concept of distance or size for its elements.

## 2. Why are normed division algebras important in geometry?

Normed division algebras are important in geometry because they provide a framework for studying geometric concepts in a more general and abstract manner. They also have applications in physics, particularly in the study of rotation and reflection symmetries.

## 3. What is the geometrical limitation of normed division algebras?

The geometrical limitation of normed division algebras is that they are only defined in dimensions that are powers of 2 (1, 2, 4, 8, etc.). This is known as the Frobenius theorem, which states that a normed division algebra can only exist in dimensions that are powers of 2.

## 4. How do normed division algebras differ from other types of algebras?

Normed division algebras differ from other types of algebras in that they have a concept of size or norm for their elements, and they also have well-defined multiplication and division operations. This sets them apart from other algebras, such as commutative algebras, where multiplication is not always defined.

## 5. What are some examples of normed division algebras?

The most well-known examples of normed division algebras are the real numbers, complex numbers, quaternions, and octonions. These are all finite-dimensional and have a norm defined on their elements. There are also infinite-dimensional normed division algebras, such as the Banach algebras and C*-algebras.

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