Have all the finite-dimensional division algebras over the reals been discovered/classified?
The are many layman accessible sources on the web describing different properties of such algebras, but all the ones I have come across seem to stop short of fully restricting them to $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$, yet do not mention the existence of anything beyond those four.
The wikipedia page on division algebras mentions that any finite-dimensional division algebra over the reals must be of dimension 1, 2, 4, or 8.
It also mentions the only finite-dimensional division algebras over the real numbers which are alternative algebras are the real numbers themselves, the complex numbers, the quaternions, and the octonions. (And this claims we don't even need the finite-dimensional qualifier for the last statement.)
Hurwitz's theorem tells us that these are also the only normed unital division algebras over the reals.
So any finite dimensional division algebra over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$ cannot have a norm if it is unital, nor have a matrix representation, nor even be alternative. Are there any known examples? Have all the possibilities been classified?
Best Answer
Real division algebras have not been classified. The best known result is the classification of flexible division algebras, which is found in this paper by Darpo. Some more details can be found in this survey paper (also by Darpo), where he states that the general classification is not known.