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:

$R$ (the real numbers)
$C$ (the complex numbers)
$H$ (the quaternions).

These algebras have dimensions $1, 2$ , 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

We identify the real multiples of $1$ with $R$ .

When we write $a \leq 0$ 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:

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

Note that if

z ∈ C\R

then

Q (z; x)

is irreducible over

R

.

The claim

The key to the argument is the following

Claim. The set Template:Mvar of all elements Template:Mvar of Template:Mvar such that

a 2 \leq 0

is a vector subspace of Template:Mvar of codimension

1

. Moreover

D = R \oplus 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

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} \backslash \mathbf {R} .

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

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

Since $zj ∈ C\R$ , the polynomials $Q (z j; x)$ 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 $a - t i = 0$ for some Template:Mvar or that $Q (z j; a) = 0$ for some Template:Mvar. The first case implies that Template:Mvar is real. In the second case, it follows that $Q (z j; x)$ 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

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 Template:Mvar the coefficient of $x 2 k -1$ 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 $Re(z j) = 0$ , in other words $tr(a) = 0$ if and only if $a 2 = -| z j | 2 < 0$ .

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.

The finish

For $a, b$ in Template:Mvar define $B (a, b) = (- ab - ba)/2$ . Because of the identity $(a + b) 2 - a 2 - b 2 = ab + ba$ , it follows that $B (a, b)$ is real. Furthermore since $a 2 \leq 0$ , we have: $B (a, a) > 0$ for $a \neq 0$ . 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 $e 1, ..., e n$ be an orthonormal basis of Template:Mvar. With respect to the negative definite bilinear form $- B$ these elements satisfy the following relations:

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

If $n = 0$ , then Template:Mvar is isomorphic to $R$ .

If $n = 1$ , then Template:Mvar is generated by $1$ and $e 1$ subject to the relation $e Template:Su = -1$ . Hence it is isomorphic to $C$ .

If $n = 2$ , it has been shown above that Template:Mvar is generated by $1, e 1, e 2$ subject to the relations

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 $n > 2$ , then Template:Mvar cannot be a division algebra. Assume that $n > 2$ . Let $u = e 1 e 2 e n$ . It is easy to see that $u 2 = 1$ (this only works if $n > 2$ ). If Template:Mvar were a division algebra, $0 = u 2 - 1 = (u - 1)(u + 1)$ implies $u = \pm1$ , which in turn means: $e n = \mp e 1 e 2$ and so $e 1, ..., e n -1$ generate Template:Mvar. This contradicts the minimality of Template:Mvar.

Remarks and related results

The fact that Template:Mvar is generated by $e 1, ..., e n$ subject to the above relations means that Template:Mvar is the Clifford algebra of $R n$ . The last step shows that the only real Clifford algebras which are division algebras are $Cℓ 0, Cℓ 1$ and $Cℓ 2$ .

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 $H$ .

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.