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:
* (the real numbers)
* (the complex numbers)
* (the quaternions).
These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but 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 be the division algebra in question.
* Let be the dimension of .
* We identify the real multiples of with .
* When we write for an element of , we tacitly assume that is contained in .
* We can consider as a finite-dimensional -vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that endomorphism. Therefore, we can speak about the trace of , and its characteristic and minimal polynomials.
* For any in define the following real quadratic polynomial:
::Q(z; x) = x^2 - 2\operatorname(z)x + , z, ^2 = (x-z)(x-\overline) \in \mathbf
:Note that if then is irreducible over .

The claim

The key to the argument is the following

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

Proof of Claim: Let be the dimension of as an -vector space, and pick in with characteristic polynomial . By the fundamental theorem of algebra, we can write

:$p(x)= (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline) \cdots (x-z_s)(x - \overline), \qquad t_i \in \mathbf, \quad z_j \in \mathbf \backslash \mathbf.$

We can rewrite in terms of the polynomials :

:$p(x)= (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x).$

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

:$p(x)= Q(z_j; x)^k = \left (x^2 - 2\operatorname(z_j) x + , z_j, ^2 \right )^k$

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

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

The finish

For in define . Because of the identity , it follows that is real. Furthermore, since , we have: for . Thus is a positive definite symmetric bilinear form, in other words, an inner product on .

Let be a subspace of that generates as an algebra and which is minimal with respect to this property. Let be an orthonormal basis of with respect to . Then orthonormality implies that:

:$e_i^2 =-1, \quad e_i e_j = - e_j e_i.$

If , then is isomorphic to .

If , then is generated by and subject to the relation . Hence it is isomorphic to .

If , it has been shown above that is generated by 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 .

If , then cannot be a division algebra. Assume that . Let . It is easy to see that (this only works if ). If were a division algebra, implies , which in turn means: and so generate . This contradicts the minimality of .

Remarks and related results

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

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.

Algebras
Quaternions
Theorems about algebras
Articles containing proofs