Frobenius theorem

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 dimensions 1, 2, and 4, respectively. Of these three algebras, the real and complex numbers are commutative, but the quaternions are not.

This theorem is closely related to Hurwitz's theorem, which states that the only normed division algebras over the real numbers are R, C, H, and the (non-associative) algebra O of octonions.

Proof

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

We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplication and we will identify d with that endomorphism. Therefore we can speak about the trace of d, the characteristic and minimal polynomials. Also, we identify the real multiples of 1 with R. When we write for an element a of D, we tacitly assume that a is contained in R. The key to the argument is the following

Claim: The set V of all elements a of D such that is a vector subspace of D of codimension 1.

To see that, we pick an a ∈ D. Let m be the dimension of D as an R-vector space. Let p(x) be the characteristic polynomial of a. By the fundamental theorem of algebra, we can write

for some real t_i and (non-real) complex numbers z_j. We have 2s + t = m. The polynomials are irreducible over R. By the Cayley-Hamilton Theorem, p(a)=0 and because D is a division algebra, it follows that either a − t_i = 0 for some i or that , z = z_j for some j. The first case implies that a ∈ R. In the second case, it follows that 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

and m=2k. The coefficient of x^{2k − 1} in p(x) is the trace of a (up to sign). Therefore we read from that equation: the trace of a is zero if and only if , that is a² = − | z | ².

Therefore V is the subset of all a with tr a = 0. In particular, it is a vector subspace (!). Moreover, V has codimension 1 since it is the kernel of a (nonzero) linear form. Also note that D is the direct sum of R and V (as vector spaces). Therefore, V generates D as an algebra.

Define now for Because of the identity (a + b)² − a² − b² = ab + ba, it follows that B(a,b) is real and since if . Thus B is a positive definite symmetric bilinear form, in other words, an inner product on V.

Let W be a subspace of V which generated D as an algebra and which is minimal with respect to that property. Let e₁,...,e_n be an orthonormal basis of W. These elements satisfy the following relations:

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

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

If n=2, we have seen above above that D is generated by 1,e₁,e₂ subject to the relations and (e₁e₂)(e₁e₂) = − 1. These are precisely the relations for H.

If n > 2, the D cannot be a division algebra! Assume that n > 2. Put u: = e₁e₂e_n. It is easy to see that u² = 1 (this only works if n > 2). Therefore 0 = u² − 1 = (u − 1)(u + 1) implies that u= ±1 (because D is still assumed to be a division algebra). But if u= ±1, then and so e₁,e₂,...e_{n − 1} generates D. This contradicts the minimality of W.

Remark: The fact that D is generated by e₁,...e_n subject to above relation can be interpreted as the statement 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.

Remark: 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 division algebra over C is C itself.

Pontryagin variant

If D is a connected, locally compact division ring, then either D = R, or D = C, or D = H.

References

• Ferdinand Georg Frobenius (1878) "Ueber 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].
• R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366-8.
• Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.