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 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 $a \leq 0$ 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:

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 \in C ∖ 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

a 2 \leq 0

is a vector subspace of

D

of dimension

n - 1

. Moreover

D = R \oplus V

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

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 $Q (z; x)$ :

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

Since $z j \in C ∖ R$ , the polynomials $Q (z j; x)$ are all irreducible over $R$ . By the Cayley–Hamilton theorem, $p (a) = 0$ and because $D$ is a division algebra, it follows that either $a - t i = 0$ for some $i$ or that $Q (z j; a) = 0$ for some $j$ . The first case implies that $a$ is real. In the second case, it follows that $Q (z j; x)$ 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

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

for some $k$ . Since $p (x)$ is the characteristic polynomial of $a$ 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 $V$ 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 $\operatorname {tr} :D\to \mathbf {R}$ . Since $R$ and $V$ are disjoint (i.e. they satisfy $\mathbf {R} \cap V=\{0\}$ ), and their dimensions sum to $n$ , we have that $D = R \oplus V$ .

The finish

For $a, b$ in $V$ 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 $B$ 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 $e 1, ..., e k$ be an orthonormal basis of $W$ with respect to $B$ . Then orthonormality implies that:

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

The form of $D$ then depends on $k$ :

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

If $k = 1$ , then $D$ is generated by $1$ and $e 1$ subject to the relation $e 21 = -1$ . Hence it is isomorphic to $C$ .

If $k = 2$ , it has been shown above that $D$ 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 $k > 2$ , then $D$ cannot be a division algebra. Assume that $k > 2$ . Define $u = e 1 e 2 e k$ and consider $u 2 =(e 1 e 2 e k)*(e 1 e 2 e k)$ . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that $u 2 = 1$ . If $D$ were a division algebra, $0 = u 2 - 1 = (u - 1)(u + 1)$ implies $u = \pm1$ , which in turn means: $e k = \mp e 1 e 2$ and so $e 1, ..., e k -1$ generate $D$ . This contradicts the minimality of $W$ .

Remarks and related results

The fact that $D$ is generated by $e 1, ..., e k$ subject to the above relations means that $D$ 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 $R$ .
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.