Talk:Normed division algebra

From Wikipedia, the free encyclopedia

The following was the original article.

A division algebra that is also a normed vector space.

The normed division algebras are:

These are the only possible normed division algebras.

The entry is clearly false as it stands, but it seems to have some hidden core of truth. How can we rescue the article? --AxelBoldt

(This is a belated response, as I didn't see this article in Recent Changes.) This result is given in John Baez's octonion article. I've never checked it myself, but I believe it's correct if you use the definition of "normed division algebra" that Baez uses: a finite-dimensional unital algebra over R that is also a normed vector space satisfying ||ab|| = ||a||.||b||. This definition appears unreasonably restrictive in disallowing infinite-dimensional algebras and algebras over C. It also conflicts with your definition of division algebra, which requires associativity. I'm not sure what to do about the article. --Zundark, 2002 Jan 9

Would it be better to omit associativity from our definition of division algebra? After all, we already have division ring. --AxelBoldt

Yes, I think we should do that. But I'm not sure what the definition should be - there are inequivalent definitions that reduce to our current definition in the associative case. I think "algebra in which the nonzero elements form a loop" is probably what we want, but it would be a good idea to check some books first to see how different authors define it. --Zundark, 2002 Jan 10

I don't claim to understand what you are talking about, but if different authors use these terms with different definitions, we should note that in the article, even if we decide to adopt one of those definitions for Wikipedia's own use. -- SJK

Yes, I usually try to do this if there is differing usage. --Zundark, 2002 Jan 10

I'm using a book at the moment, On Quaternions and Octonions, which uses the term composition algebra in the proof of Hurwitz's theorem, which seems to be normally stated for normed division algebras - so I'm editting the definition to have this as an alternative name, and I'll re-direct an article called Composition Algebra to this page. (MatthewMain 20:47, 29 April 2006 (UTC))

I'm sorry, there is now an article called Composition Algebra and has been for 3 months! Serves me right for not checking really... (MatthewMain 20:54, 29 April 2006 (UTC))

[edit] Finite/infinite

Can someone confirm (with cite) that there are no infinte NDAs over the reals? Then we can change the William Rowan Hamilton article to reflect that. Rich Farmbrough, 12:41 12 October 2006 (GMT).