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

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:
${\displaystyle Q(z;x)=x^{2}-2\operatorname {Re} (z)x+|z|^{2}=(x-z)(x-{\overline {z}})\in \mathbf {R} [x].}$
Note that if zC ∖ 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 = RV 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

${\displaystyle p(x)=(x-t_{1})\cdots (x-t_{r})(x-z_{1})(x-{\overline {z_{1}}})\cdots (x-z_{s})(x-{\overline {z_{s}}}),\qquad t_{i}\in \mathbf {R} ,\quad z_{j}\in \mathbf {C} \setminus \mathbf {R} .}$

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

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

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

${\displaystyle p(x)=Q(z_{j};x)^{k}=\left(x^{2}-2\operatorname {Re} (z_{j})x+|z_{j}|^{2}\right)^{k}}$

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 ${\displaystyle \operatorname {tr} :D\to \mathbf {R} }$. Since R and V are disjoint (i.e. they satisfy ${\displaystyle \mathbf {R} \cap V=\{0\}}$), and their dimensions sum to n, we have that D = RV.

The finish

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 B. Then orthonormality implies that:

${\displaystyle e_{i}^{2}=-1,\quad e_{i}e_{j}=-e_{j}e_{i}.}$

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

If k = 1, then D is generated by 1 and e1 subject to the relation e2
1
= −1
. Hence it is isomorphic to C.

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

${\displaystyle e_{1}^{2}=e_{2}^{2}=-1,\quad e_{1}e_{2}=-e_{2}e_{1},\quad (e_{1}e_{2})(e_{1}e_{2})=-1.}$

These are precisely the relations for H.

If k > 2, then D cannot be a division algebra. Assume that k > 2. Define u = e1e2ek and consider u2=(e1e2ek)*(e1e2ek). By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that u2 = 1. If D were a division algebra, 0 = u2 − 1 = (u − 1)(u + 1) implies u = ±1, which in turn means: ek = ∓e1e2 and so e1, ..., ek−1 generate D. This contradicts the minimality of W.

Remarks and related results

• The fact that D is generated by e1, ..., ek 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.