Frobenius theorem (real division algebras)

Template:Short description Script error: No such module "For". 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 real dimension Template:Math, and Template:Math, respectively. Of these three algebras, Template:Math and Template:Math are commutative, but Template:Math is not.

Proof

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

Let Template:Math be the division algebra in question.
Let Template:Math be the dimension of Template:Math.
We identify the real multiples of Template:Math with Template:Math.
When we write Template:Math for an element Template:Mvar of Template:Mvar, we imply that Template:Mvar is contained in Template:Math.
We can consider Template:Mvar as a finite-dimensional Template:Math-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 Template:Math define the following real quadratic polynomial:

Q (z; x) = x^{2} - 2 Re (z) x + | z |^{2} = (x - z) (x - \overline{z}) \in 𝐑 [x] .

The key to the argument is the following

Claim. The set Template:Mvar of all elements Template:Mvar of Template:Mvar such that Template:Math is a vector subspace of Template:Mvar of dimension Template:Math. Moreover Template:Math as Template:Math-vector spaces, which implies that Template:Mvar generates Template:Mvar as an algebra.

Proof of Claim: Pick Template:Mvar in Template:Mvar with characteristic polynomial Template:Math. By the fundamental theorem of algebra, we can write

p (x) = (x - t_{1}) \dots (x - t_{r}) (x - z_{1}) (x - \overline{z_{1}}) \dots (x - z_{s}) (x - \overline{z_{s}}), t_{i} \in 𝐑, z_{j} \in 𝐂 ∖ 𝐑 .

We can rewrite Template:Math in terms of the polynomials Template:Math:

p (x) = (x - t_{1}) \dots (x - t_{r}) Q (z_{1}; x) \dots Q (z_{s}; x) .

Since Template:Math, the polynomials Template:Math are all irreducible over Template:Math. By the Cayley–Hamilton theorem, Template:Math and because Template:Mvar is a division algebra, it follows that either Template:Math for some Template:Mvar or that Template:Math for some Template:Mvar. The first case implies that Template:Mvar is real. In the second case, it follows that Template:Math is the minimal polynomial of Template:Mvar. Because Template:Math has the same complex roots as the minimal polynomial and because it is real it follows that

p (x) = Q (z_{j}; x)^{k} = {(x^{2} - 2 Re (z_{j}) x + | z_{j} |^{2})}^{k}

for some Template:Math. Since Template:Math is the characteristic polynomial of Template:Mvar the coefficient of Template:Math in Template:Math is Template:Math up to a sign. Therefore, we read from the above equation we have: Template:Math if and only if Template:Math, in other words Template:Math if and only if Template:Math.

So Template:Mvar is the subset of all Template:Mvar with Template:Math. In particular, it is a vector subspace. The rank–nullity theorem then implies that Template:Mvar has dimension Template:Math since it is the kernel of $tr : D \to 𝐑$ . Since Template:Math and Template:Mvar are disjoint (i.e. they satisfy $𝐑 \cap V = {0}$ ), and their dimensions sum to Template:Mvar, we have that Template:Math.

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 Template:Math be an orthonormal basis of Template:Mvar with respect to Template:Math. Then orthonormality implies that:

e_{i}^{2} = - 1, e_{i} e_{j} = - e_{j} e_{i} .

The form of Template:Mvar then depends on Template:Mvar:

If Template:Math, then Template:Mvar is generated by Template:Math and Template:Math subject to the relation Template:Math. Hence it is isomorphic to Template:Math.

If Template:Math, it has been shown above that Template:Mvar is generated by Template:Math subject to the relations

e_{1}^{2} = e_{2}^{2} = - 1, e_{1} e_{2} = - e_{2} e_{1}, (e_{1} e_{2}) (e_{1} e_{2}) = - 1 .

These are precisely the relations for Template:Math.

If Template:Math, then Template:Mvar cannot be a division algebra. Assume that Template:Math. Define Template:Math and consider Template:Math. By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that Template:Math. If Template:Mvar were a division algebra, Template:Math implies Template:Math, which in turn means: Template:Math and so Template:Math generate Template:Mvar. This contradicts the minimality of Template:Mvar.

The fact that Template:Mvar is generated by Template:Math subject to the above relations means that Template:Mvar is the Clifford algebra of Template:Math. The last step shows that the only real Clifford algebras which are division algebras are Template:Math and Template:Math.
As a consequence, the only commutative division algebras are Template:Math and Template:Math. Also note that Template:Math is not a Template:Math-algebra. If it were, then the center of Template:Math has to contain Template:Math, but the center of Template:Math is Template:Math.
This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are Template:Math, and the (non-associative) algebra Template:Math.
Pontryagin variant. If Template:Mvar is a connected, locally compact division ring, then Template:Math, or Template:Math.

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 Template:ISBN .
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.