Frobenius theorem (real division algebras)

Short description: Theorem in abstract algebra

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:

* R (the real numbers)
* C (the complex numbers)
* H (the quaternions).

These algebras have real dimension 1, 2, and 4, respectively. Of these three algebras, R and C are commutative, but H is not.

## Contents

* 1 Proof
* 1.1 Introducing some notation
* 1.2 The claim
* 1.3 The finish
* 2 Remarks and related results
* 3 References

## Proof

The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.

### Introducing some notation

* Let D be the division algebra in question.
* Let n be the dimension of D.
* We identify the real multiples of 1 with R.
* When we write a ≤ 0 for an element a of D, we imply that a is contained in R.
* We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplication, we identify d with that endomorphism. Therefore, we can speak about the trace of d, and its characteristic\- and minimal polynomials.
* For any z in C define the following real quadratic polynomial:

Q(z;x)=x2−2Re⁡(z)x+|z|2=(x−z)(x−z‾)∈𝐑[x].
Note that if z ∈ C ∖ R then Q(z; x) is irreducible over R.

### The claim

The key to the argument is the following

Claim. The set V of all elements a of D such that a2 ≤ 0 is a vector subspace of D of dimension n − 1. Moreover D = R ⊕ V as R-vector spaces, which implies that V generates D as an algebra.

Proof of Claim: Pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we can write

p(x)=(x−t1)⋯(x−tr)(x−z1)(x−z1‾)⋯(x−zs)(x−zs‾),ti∈𝐑,zj∈𝐂∖𝐑.

We can rewrite p(x) in terms of the polynomials Q(z; x):

p(x)=(x−t1)⋯(x−tr)Q(z1;x)⋯Q(zs;x).

Since zj ∈ C ∖ R, the polynomials Q(zj; x) are all irreducible over R. By the Cayley–Hamilton theorem, p(a) = 0 and because D is a division algebra, it follows that either a − ti = 0 for some i or that Q(zj; a) = 0 for some j. The first case implies that a is real. In the second case, it follows that Q(zj; x) is the minimal polynomial of a. Because p(x) has the same complex roots as the minimal polynomial and because it is real it follows that

p(x)=Q(zj;x)k=(x2−2Re⁡(zj)x+|zj|2)k

Since p(x) is the characteristic polynomial of a the coefficient of x 2k − 1 in p(x) is tr(a) up to a sign. Therefore, we read from the above equation we have: tr(a) = 0 if and only if Re(zj) = 0, in other words tr(a) = 0 if and only if a2 = −|zj|2 < 0.

So V is the subset of all a with tr(a) = 0. In particular, it is a vector subspace. The rank–nullity theorem then implies that V has dimension n − 1 since it is the kernel of tr⁡:D->𝐑. Since R and V are disjoint (i.e. they satisfy 𝐑∩V={0}), and their dimensions sum to n, we have that D = R ⊕ V.

### The finish

For a, b in V define B(a, b) = (−ab − ba)/2. Because of the identity (a \+ b)2 − a2 − b2 = ab \+ ba, it follows that B(a, b) is real. Furthermore, since a2 ≤ 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V.

Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property. Let e1, ..., en be an orthonormal basis of W with respect to B. Then orthonormality implies that:

ei2=−1,eiej=−ejei.

If n = 0, then D is isomorphic to R.

If n = 1, then D is generated by 1 and e1 subject to the relation e21 = −1. Hence it is isomorphic to C.

If n = 2, it has been shown above that D is generated by 1, e1, e2 subject to the relations

e12=e22=−1,e1e2=−e2e1,(e1e2)(e1e2)=−1.

These are precisely the relations for H.

If n > 2, then D cannot be a division algebra. Assume that n > 2. Let u = e1e2en. It is easy to see that u2 = 1 (this only works if n > 2). If D were a division algebra, 0 = u2 − 1 = (u − 1)(u \+ 1) implies u = ±1, which in turn means: en = ∓e1e2 and so e1, ..., en−1 generate D. This contradicts the minimality of W.

## Remarks and related results

* The fact that D is generated by e1, ..., en subject to the above relations means that D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are division algebras are Cℓ0, Cℓ1 and Cℓ2.
* As a consequence, the only commutative division algebras are R and C. Also note that H is not a C-algebra. If it were, then the center of H has to contain C, but the center of H is R. Therefore, the only finite-dimensional division algebra over C is C itself.
* This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are R, C, H, and the (non-associative) algebra O.
* Pontryagin variant. If D is a connected, locally compact division ring, then D = R, C, or H.

## References

* Ray E. Artz (2009) Scalar Algebras and Quaternions, Theorem 7.1 "Frobenius Classification", page 26.
* Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405.
* Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp. 30–2 ISBN 0-7923-2459-5 .
* Leonard Dickson (1914) Linear Algebras, Cambridge University Press . See §11 "Algebra of real quaternions; its unique place among algebras", pages 10 to 12.
* R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8.
* Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.

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