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. The theorem proves that the only associative division algebra which is not commutative over the real numbers is the quaternions.

If D is a finite dimensional division algebra over the real numbers R then one of the following cases holds

• D = R
• D = C (complex numbers)
• D = H (quaternions).

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 $a\leq 0$ 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 $a^{2}\leq 0$ is a subvector space 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

$p(x)=(x-t_{1})...(x-t_{r})(x-z_{1})(x-{\overline {z_{1}}})...(x-z_{s})(x-{\overline {z_{s}}})$

for some real t_i and (nonreal) complex numbers z_j. We have 2s+t=m. The polynomials $x^{2}-2Re(z_{i})+|z_{i}|^{2}=(x-z_{i})(x-{\overline {z_{i}}})$ 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 $a^{2}-2Reza+|z|^{2}=0$, $z=z_{j}$ for some j. The first case implies that a ∈ R. In the second case, it follows that $x^{2}-2Rezx+|z|^{2}$ 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

$p(x)=(x^{2}-2Rezx+|z|^{2})^{k}$

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 $Re(z)=0$, that is $a^{2}=-|z|^{2}$.

Therefore V is the subset of all a with tr (a)=0. In particular, it is a sub-vector space (!). 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 a,b ∈ V: B(a,b):= - ab -ba. Because of the identity $(a+b)^{2}-a^{2}-b^{2}=ab+ba$, it follows that B(a,b) is real and since $a^{2}\leq 0,B(a,a)>0$ if $a\neq 0$. Thus B is a positive definite symmetric bilinear form, in other words, an inner product on V.

Let W be a subspace of V which generated D as an algebra and which is minimal with respect to that property. Let $e_{1},...,e_{n}$ be an orthonormal basis of W. The elements e_i satisfy the following relations:

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

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

If n=1, then D is generated by 1 and e_1 subject to the relation e_1^2=-1. Hence it is isomorphic to C

If n = 2, we have seen above above thatD is generated by $1,e_{1}$ and $e_{2}$ subject to the relations $e_{1}^{2}=e_{2}^{2}=-1$, $e_{1}e_{2}=-e_{1}e_{2}$ and $(e_{1}e_{2})(e_{1}e_{2})=-1$. These are precisely the relations for H.

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

Remark: The fact that D is generated by $e_{1},...e_{n}$ subject to above relation can be interpreted as the statement that D is the Clifford algebra of R^n. The last step shows that the only Clifford algebras which are division algebras are Cl^1, Cl^2 and Cl^3.

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.