Talk:Para-quaternion

Relation with coquaternions
The structure identified in this article is the same as the coquaternion ring originated and named by James Cockle. No reference is given for the isomorphism asserted. The reference linked is too technical to be of use by the non-expert. The assertion of use of para-quaternions in string theory is not backed-up. Should this page be retained in Wikipedia ?
 * Rgdboer 23:58, 21 January 2007 (UTC)

There are 69 papers about para-quaternion geometry in arxiv.org (as today's Google search shows). A good number of these in hep-th and math-ph. I think this notion is notable enough to be included. There is not a single mention of co-quaternions in any form ( [ 1 | 2 3 ] ), which might explain why I did not mention any.

The correct formula for para-quaternions (pluses, minuses) is very easy to forget, hence I need this article. I suppose same is true for other geometers.

Best regards, Tiphareth 18:43, 22 January 2007 (UTC)


 * It looks like we have yet another candidate for the same thing ... Oh my ... Let's hope Wikipedia will at some time have a good reference across all those terms. Contained in Lie and Clifford algebra, I vote for merging all the split-/para-/hyperbolic-/co-quaternions into Coquaternion, and there make a section that lists, in the order of their re-invention, the isomorphism and potential differences in their definition. I believe that would be convincing encyclopedic content. Thanks, Jens Koeplinger 01:45, 20 May 2007 (UTC)


 * Agree. As long as this section is clear and easy to find, it's the best. Right now the Coquaternion article is overloaded with content, and I don't have the editing skills required to make this merge cleanly and succintly. And if the merge is done in a careless way, it will destroy the value of all merged articles. Tiphareth 15:05, 20 May 2007 (UTC)


 * Uh oh, I see work coming my way; it's my declared interest in making the many different algebras more easily accessible In Wikipedia, and it might be time now to live-up to my promises :) ... It's a terminology mess out there, but general popularity of algebras appears to be on the rise, and I want to make sure we represent in Wikipedia the lines between what's already invented, what's serious, what's useful, and what is pseudoscience ... In the next weeks I'll look at some well-written articles, how they're structured, and try to come up with an improvement to several algebra-related articles we already have. We've gotten much good additional content recently from Jheald, and the only thing I'd like to have is relations of some of these algebras to Lie algebras. I'll go and ask around who may be interested in providing isomorphisms or classifications (any help is appreciated). Thanks, Jens Koeplinger 15:33, 20 May 2007 (UTC)


 * Great! Keep in mind that at some point the articles on para-quaternionic geometry (manifolds and related structures) will be added, so the links to the algebra and its relations should remain accessible. Regarding the hypercomplex algebra set-up, I think the Clifford algebra approach is the clearest and most up-to-date. Here are the links that I found very useful: 1, 2, 3. Also, please keep this talk page for history. Tiphareth 16:57, 20 May 2007 (UTC)


 * I've started merging this material into split-quaternion. I agree that the latter needs much rewriting, and have commented on the Lie theory on the talk page. I will need the help promised above to complete the merge in a form which everyone will find satisfactory. This page will become a redirect, and hence its talk page history will be preserved. Geometry guy 12:17, 8 June 2007 (UTC)

I added the link to coquaterions and explained the isomorphism with the matrix algebra. This is of course the same notion, put in a different context (and Cl(1,1) is another guise of the same algebraic object).

It's a pity that most of people working in para-quaternionic geometry don't realize this is James Cockle's invention. In fact, I suppose most of them did not know about Cockle - I did not, until you pointed out. Thanks! Tiphareth 19:50, 22 January 2007 (UTC)

Good to see the links to Cockle and coquaternions. For the real matrix isomorphism, rather than depend on the Clifford algebra classification, I have inserted the keys to a direct isomorphism at coquaternion. Perhaps you could explain "manifold with paraquaternionic structure". A note or link to the feature of string theory application would also be nice. As for the comparative volume of external links, I am not persuaded that such material trumps the historical roots of the concept as given in coquaternions notes and references. Given the experience of split-complex number alternative names, we can expect to find yet more monikers for this object.
 * Rgdboer 22:08, 27 January 2007 (UTC)

Thanks. I added a couple of links to the literature (both physics and mathematics) and also added the reference to explicit isomorphism with Mat(2,\R) from co-quaternions. Regarding the para-quaternionic manifolds, this is too big a subject to touch here - another article should be added (and I am not qualified, I'm afraid, there is too much stuff I did not read: the field is way too active). Tiphareth 16:11, 3 March 2007 (UTC)

Merge tag added
I've added a merge tag. There really ought only to be one article covering all uses of this object, not two. Jheald 20:06, 29 March 2007 (UTC)

I don't think it's wise. The "co-quaternions" belong to the history, this notion is not used now. The coquaternion article is very faithful to this theme. There are hundred of papers on paraquaternion geometry, which have almost no relation to the content of past research in low-dimensional algebra.

The only use of Para-quaternion article is to have a sign convention fixed in one place. Tiphareth 07:47, 30 March 2007 (UTC)


 * Agree with Jheald. We cannot have two articles on the same topic and there now seems to be agreement on this in the section above. I have merged the information in this article into split-quaternion with the intention of replacing it by a redirect. If the emphasis in the latter article is unsatisfactory, the solution is to update it, not to create a separate article. Geometry guy 12:05, 8 June 2007 (UTC)

Sign convention
One curiosity:

The paraquaternions are defined with ijk = -1, so k= -ij.

The coquaternions are defined with k = +ij, so ijk = +1.

It still works out to be the same algebra, just k has had a sign flip. Jheald 20:13, 29 March 2007 (UTC)