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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 , 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 imply 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:
:
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: Pick in with characteristic polynomial . By the fundamental theorem of algebra, we can write

We can rewrite in terms of the polynomials :

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

for some . 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 . Since and are disjoint (i.e. they satisfy ), 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:

The form of then depends on :

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

These are precisely the relations for .

If , then cannot be a division algebra. Assume that . Define and consider . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that . 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 .
  • 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 .

See also

References