### New Normed Division Algebra Found!

#### Posted by John Baez

Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known. I’ve spent a lot of time studying them.

Thus you can imagine my surprise today when I learned Hurwitz’s theorem was false!

- Joy Christian, Eight-dimensional octonion-like but associative normed division algebra,
*Communications in Algebra*(2020), 1-10.

Abstract.We present an eight-dimensional even sub-algebra of the $2^4=16$-dimensional associative Clifford algebra $\mathrm{Cl}_{4,0}$ and show that its eight-dimensional elements denoted as $\mathbf{X}$ and $\mathbf{Y}$ respect the norm relation $\| \mathbf{X} \mathbf{Y}\| = \| \mathbf{X} \| \| \mathbf{Y} \|$, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Even more wonderful is that the author has discovered that the unit vectors in his normed division algebra form a 7-sphere that is not homeomorphic to the standard 7-sphere. Exotic 7-spheres are a dime a dozen, but those merely fail to be *diffeomorphic* to the standard 7-sphere.

Well, no: this article must be wrong.

Doesn’t *Communications in Algebra* have some mechanism where a few editors look at every paper — just to be sure they make sense?

## Re: New Normed Division Algebra Found!

John,

I cannot match your knowledge in mathematics as I am merely a physicist (you may even remember me from Perimeter Institute or from one of the meetings at Roger Penrose’s house in Oxford). But I believe that if you actually read my paper you might find that it is not all that wrong, misguided, or uninteresting. It might even give you a new perspective — a Geometric Algebra perspective. On the other hand, if you find a mistake in my calculations, then please do let me know. I am always eager to learn.

Joy