Frobenius theorem (real division algebras)

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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:

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

Proof[edit]

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

Introducing some notation[edit]

  • Let D be the division algebra in question.
  • We identify the real multiples of 1 with R.
  • When we write a ≤ 0 for an element a of D, we tacitly assume 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:
Note that if zC\R then Q(z; x) is irreducible over R.

The claim[edit]

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 codimension 1. Moreover D = RV as R-vector spaces, which implies that V generates D as an algebra.

Proof of Claim: Let m be the dimension of D as an R-vector space, and pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we can write

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

Since zjC\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 ati = 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

Since p(x) is the characteristic polynomial of a the coefficient of x2k−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. Moreover, V has codimension 1 since it is the kernel of a non-zero linear form, and note that D is the direct sum of R and V as vector spaces.

The finish[edit]

For a, b in V define B(a, b) = (−abba)/2. Because of the identity (a + b)2a2b2 = 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 the negative definite bilinear form B 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 e2
1
= −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

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[edit]

  • 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 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[edit]