Frobenius theorem (real division algebras)
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.
Introducing some notation
- We identify the real multiples of 1 with R.
- When we write a ≤ 0 for an element Template:Mvar of Template:Mvar, we tacitly assume that Template:Mvar is contained in R.
- We can consider Template:Mvar as a finite-dimensional R-vector space. Any element Template:Mvar of Template:Mvar defines an endomorphism of Template:Mvar by left-multiplication, we identify Template:Mvar with that endomorphism. Therefore we can speak about the trace of Template:Mvar, and its characteristic and minimal polynomials.
- For any Template:Mvar in C define the following real quadratic polynomial:
The key to the argument is the following
- Claim. The set Template:Mvar of all elements Template:Mvar of Template:Mvar such that a2 ≤ 0 is a vector subspace of Template:Mvar of codimension 1. Moreover D = R ⊕ V as R-vector spaces, which implies that Template:Mvar generated Template:Mvar as an algebra.
Proof of Claim: Let Template:Mvar be the dimension of Template:Mvar as an R-vector space, and pick Template:Mvar in Template:Mvar 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 zj ∈ C\R, the polynomials Q(zj; x) are all irreducible over R. By the Cayley–Hamilton theorem, p(a) = 0 and because Template:Mvar is a division algebra, it follows that either a − ti = 0 for some Template:Mvar or that Q(zj; a) = 0 for some Template:Mvar. The first case implies that Template:Mvar is real. In the second case, it follows that Q(zj; x) is the minimal polynomial of Template:Mvar. 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 Template:Mvar 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 Template:Mvar is the subset of all Template:Mvar with tr(a) = 0. In particular, it is a vector subspace. Moreover, Template:Mvar has codimension 1 since it is the kernel of a non-zero linear form, and note that Template:Mvar is the direct sum of R and Template:Mvar as vector spaces.
For a, b in Template:Mvar 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 Template:Mvar is a positive definite symmetric bilinear form, in other words, an inner product on Template:Mvar.
Let Template:Mvar be a subspace of Template:Mvar that generates Template:Mvar as an algebra and which is minimal with respect to this property. Let e1, ..., en be an orthonormal basis of Template:Mvar. With respect to the negative definite bilinear form −B these elements satisfy the following relations:
If n = 2, it has been shown above that Template:Mvar is generated by 1, e1, e2 subject to the relations
These are precisely the relations for H.
If n > 2, then Template:Mvar 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 Template:Mvar 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 Template:Mvar. This contradicts the minimality of Template:Mvar.
- The fact that Template:Mvar is generated by e1, ..., en subject to the above relations means that Template:Mvar 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 Template:Mvar is a connected, locally compact division ring, then D = R, C, or H.
- 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.