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It is classical to take a division algebra over $\mathbb{R}$ and defining an H-space structure on the unit spheres by restricting and normalizing.

There are commutative division algebras of dimension 1 and 2 leading to commutative products on $S^0$ and $S^1$ identifying them as Eilenberg-MacLane spaces - Or if we forget some structure as an $E_{\infty}$-spaces.

The associative division algebras $\mathbb{H}$ defines an associative product on $S^3$, which is also a Lie-group, but forgetting some structure it is an $A_\infty$-space.

There division algebra $\mathbb{O}$ defines an $A_2$ structure on $S^7$, which is not $A_\infty$ (is it $A_3$?).

As is well known it is possible to prove that no other spheres has $A_2$ structure.

Question: Is there a heiraki of structures below $A_2$ yet related such that $S^{15}$ has this structure, but $S^{31}$ does not?

Remark: A heiraki below $A_2$ could be that $A_2=D_\infty$ for some definition of structures $D_n$, analagous to $E_1$ being $A_\infty$.

Question: Is there an even more general definition of "lower" structures and a statement about all spheres (including possibly non-trivial structures on even-dimensional spheres)?

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I cannot see where real division algebras enter in your questions? What is needed, really, to construct those structures on spheres is a normed division algebra structure on $\mathbb R^n$, and those exist only for $n=1,2,4,8$. – Mariano Suárez-Alvarez Mar 31 '10 at 16:33
I am interested in the product on the spheres not the division algebras, they are just very related in all non-trivial cases i know, and thus maybe also to generalizations. – Thomas Kragh Apr 1 '10 at 10:23
What happened to the comment about laxing the unital condition - I thought that sounded very promissing! – Thomas Kragh Apr 3 '10 at 10:18

I like your spelling of hierarchy! $S^7$ is not $A_3$ -- if it were, you could construct the projective space $\mathbb{O}P^3$, but that's impossible (some decomposition of Steenrod operations argument). You'll find this and answers to your other question in Baez's article on the octonions in the Bulletin of the AMS. There are higher dimensional algebras in the sequence $\mathbb R$, $\mathbb C$, $\mathbb H$, $\mathbb O$, which you can get by the Cayley-Dickson construction, but they are not normed, so you can't take the unit sphere. The 16-dimensional guy is "power-commutative", meaning that powers of an element x commute with x (not obvious if you're not associative), this starts failing in dimension 32 if I remember correctly. So there's some sort of hierarchy of structure on the algebras themselves, if not on the spheres.

Some of the spheres have an H-space structure or a loop space structure after completing at a prime p ($S^{2p-3}$, for example). Maybe this goes a bit in the direction of your question about some kind of multiplication on higher spheres.

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The precise properties which are lost as you extend from one Cayley-Dickson algebra to the next are not known. A recent paper by Biss, Dugger, and Isaksen (all Dan's!) gives some results about the dimension of the space of annihilators of any particular element. Another paper by Biss, Christensen, Dugger, and Isaksen (still all Dan's!) discusses the eigentheory of the Cayley-Dickson algebras. You can find the papers here: – Bill Kronholm Mar 31 '10 at 17:02
I have looked at the paper (very nice paper) and the 16-dimensional Cayley-Dickson construction on $\mathbb{O}$ has zero-divisors, and it thus seems (to me at least) to have no relevance to structures on $S^{15}$. – Thomas Kragh Apr 2 '10 at 9:12
The algebras that Cayley-Dickson construction produces are always power-associative, as mentioned here: – Junyan Xu Nov 25 '12 at 22:47

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