My only point was to assert that the definition of "division algebra" is well established, and to say what this is: It is your third definition below, starting with "Finally". It is interesting that you have found certain structures on R^16 that have a multiplication. But traditionally people have sought multiplications on R^n that distribute over the vector addition. —Dan
On Dec 28, 2015, at 9:51 AM, Warren D Smith <warren.wds@gmail.com> wrote:
----- Dan, I already knew what you said about real division algebras and topology. However, I do not understand those topological proofs. I in fact asked Milnor for help at one point. I have a conjecture that we ought to be able to use "generalized smoothness" (a new topological notion I introduced in this paper) to prove that the only real division algebras, (wide sense) were 1,2,34,8,16 dimensional. Milnor and another Topologist, Dennis Sullivan;, agreed this was interesting problem. However, Milnor said he no longer was able to do this stuff and no longer remembered how the topological proofs he was so famous for, work. Sullivan claimed he was going to work on it, but if so, he didn't accomplish anything, at least not that he told me about. I suggest reading my paper, especially the parts about division, e.g. sections 19-23. Here are some quotes: nnacd = not necessarily associative, commutative, or distributive. 2^n-ons: for n=1,2,3,4 these are the reals, complexes, quaternions, octonions, and my 16-ons. 19. What is a “division algebra” and are the 2^n-ons one? One (boring) definition of “division algebra” is “an algebra in which xy = 0 implies x = 0 or y = 0.” By that definition, all of our 2^n-ons are nnacd “division algebras” simply because of the multiplicativity of Euclidean norm. A stronger definition of “division algebra” is “an algebra in which, given b and given a not= 0, 1. there exists z so that az = b, and 2. there exists q so that qa = b.” Finally, a still stronger definition would require, not only that the solutions z and q of these two division problems must exist, but that, furthermore, they must be unique. . . . . . . -----