Talk:Split-quaternion

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I think split-quaternion would be a better name for this page; mostly for naming consistency with split-complex number. We could then do split-octonion as well. As you've pointed out all these structures have numerous names; we should, however, strive for some consistency. -- Fropuff 16:52, 2005 Feb 6 (UTC)

Contents 1 "Hyperbolic Quaternion" ambiguity 2 re: Clifford algebras 3 Name of this article: James Cockle's ring or something more modern? 3.1 Inner products

[edit] "Hyperbolic Quaternion" ambiguity

(19 June 2006)

I see that there is a potential ambiguity about the term "Hyperbolic quaternion". The reference I've added to the split-quaternion site refers to hyperbolic quaternions from the hypernumbers program only, and not to the hyperbolic quaternions as they were defined in the 19th century. I hope that my wording is clear and the reference is therefore not ambiguous. In addition, I've explicitely added hyperbolic quaternions to the referenced hypernumbers page (as an example), and am also referencing split-quaternions / coquaternions from there.

While I am open to any wording or placement adjustment, I see this as a correct and valid statement, and see the reference therefore as applicable, useful, and reasonable. I would be glad to discuss any concern. Hypernumbers in general are actively discussed in the "hypernumbers" Yahoo(R) group. Thanks, Jens.

I've done some layout clean-up, as a step in the right direction, hopefully. Comments are invited. Thanks, Jens Koeplinger 02:08, 19 August 2006 (UTC)

[edit] re: Clifford algebras

Since Clifford algebras are a more advanced concept than a particular 4-dimensional real algeba, the introduction of the extra concept in the introductory paragraph amounts to obscurantism. Clear expository principles should prevail over the urge to link in an already well-funded research area. The better we describe the lower-dimensional algebras, the firmer will be the foundation for higher study. So, I have removed the Clifford algebra link from the first paragraph and replaced it with a more elementary and concrete reference.Rgdboer 01:59, 30 March 2007 (UTC)

I've put it back again. The point is that identifying the Clifford algebra(s) immediately tags what this algebra is about, what it is likely to be useful for, and how it compares to other hypercomplex numbers or quaternion variants. It means that somebody who can read a Clifford algebra classification knows exactly what to expect to find in the article.

So seeing Cℓ_2,0(R), one can read off immediately that this is going to be the natural algebra for describing vectors and directed areas in 2d, and their behaviour under reflections and rotations; and co-ordinate transformations on hyperbolic surfaces in 3d through the isomorphism with Cℓ⁰_1,2(R). Seeing Cℓ_1,1(R), one can read off immediately that this is going to be a natural algebra for describing vectors and areas on surfaces with one real and one hyperbolic coordinate; and through the isomorphism with Cℓ⁰_2,1(R), it's going to relate co-ordinate transformations on hyperbolic surfaces in 3d with the alternative metric convention (+,+,-), rather than (-,-,+).

As for Clifford algebras being obscurantist, I agree that IMO the current version of the Clifford algebra article starts at much too abstract, too all-encompassing, too mathemematically sophisticated a level. But that is a problem with the current article, not with the topic. If you look at hypercomplex number, you can start to see how useful the Clifford algebra classifications can be for making sense of all these algebras, and linking them all together with a systematic contextualised approach. I'm also going to put in some work on the geometric algebra article, which has potential to become the consolidated entry-level way in to study algebras like this one, and the geometric uses to which they can be put. See for example the first couple of chapters of the Cambridge "Physical applications of geometric algebra" course, linked at Geometric_algebra#Further_reading for how it ties everything together.

If you read that link, I hope you'll agree that it ought to be much more easy for students to study all the examples of geometric algebras together, understanding their geometric uses in a systematic way, rather than trying to pick them off in an ad-hoc unconnected way one by one. Jheald 07:48, 30 March 2007 (UTC)

I side with Rgdboer, Clifford algebras are only one out of many aspects of approaching coquaternions, historically and algebraically. The current article mentions the various isomorphisms in a way that is scattered out throughout the article, which is a problem. I believe that this article needs much rework. However, I am very concerned about the current edits in the hypercomplex number article by Jheald. The article suddently becomes an article on Clifford algebras, rather than an overview article. I have voiced my concern there. Thanks, Koeplinger 19:36, 30 March 2007 (UTC)

[edit] Name of this article: James Cockle's ring or something more modern?

There is a long tradition in astronomy and mathematics that a discoverer has the rights to name his discovery. In that tradition the original title of this page is Coquaternion. Recently the page was moved to Split-quaternion on the grounds that this term is more commonly used, though no evidence was provided. Within WP there are several pages dealing with naming disputes, though the particulars of other disputes seem irrelevant here. Advice given is that a resolution through dialogue is preferable to polling. On the side of "coquaternion" one can note the twin status of this ring with Hamilton's by its appearance as real matrices (2 x 2) when re-arranged. If you happen by this talk page, click the edit button and contribute to this stand-off: discoverer's designation or "split-quaternion", what do you think? >>Rgdboer 21:22, 19 May 2007 (UTC)

My vote, for the moment anyway, would be to stick with co-quaternion, because I don't (yet) see why the characterisation of this algebra as the result of applying the Cayley-Dickson process to the algebra Cl_1,0(R) is particularly interesting. If material were to be added to the article to motivate why "splitness" is an gives useful/meaningful/interesting insight into the features of the algebra, I'd happily reconsider. But as the article stands at the moment, renaming to "split quaternion" seems not appropriate, given the balance of properties the article focusses on. Jheald 22:05, 19 May 2007 (UTC)

An apologetic entry: I am the miscreant who moved this page from coquaternion to split-quaternion. The naming of things seems to generate more discussion than anything else on wikipedia. Anyway, I don't have a particularly strong opinion, so I will try to bow out gracefully with a few comments. First of all I disagree that there is a tradition in mathematics that a discoverer names their discovery. In fact normally, someone else names it after the discoverer. Second, as far as I am aware, co-quaternion is rarely used in mathematics these days. It may be used in astronomy, but this is beyond my expertise. The "See also" section of this article indicates why the "split" terminology is gaining ground. This is associated with the idea of the split form of a semisimple Lie algebra. Finally, I have no desire to create a stand-off. If there is consensus to move this page back, then so be it. If not, I am more than willing to improve the article, add historical information on the terminology, and replace "co" by "split" where appropriate. Geometry guy 22:39, 19 May 2007 (UTC)

A note on what it is about this algebra that associates it with "split form of a semisimple Lie algebra" would be a useful addition to this article, whatever it ends up being called. I can't say it's obvious to me from any of the "see also" articles, and the idea of a "split form" doesn't seem to be mentioned in the semisimple Lie algebra. So further explanation of this concept, and hence of the name "split quaternion" would be welcome, regardless of what the article's final naming. Jheald 00:09, 20 May 2007 (UTC)

To try to answer my own question, (and to see if I can correctly interpret what G-guy wrote), I think the "split" comes in when we identify this algebra as parametrising rotations. Whereas on the one hand the quaternions can be identified with Cℓ⁰_3,0(R), ie the even Clifford sub-algebra that is associated with rotations in a space with the pure signature (+,+,+), on the other hand the split-quaternions can be identified with Cℓ⁰_2,1(R) and Cℓ⁰_1,2(R) - rotations (partially hyperbolic) in 3d pseudo-Euclidean space with the split signature (+,+,-) or (+,-,-).

This is analogous to complex numbers being identified with Cℓ⁰_2,0(R) - rotations in 2D space with the pure signature (+,+), whereas split-complex numbers are identified with Cℓ⁰_1,1(R) - hyperbolic rotations in 2D pseudo-Euclidean space with the split signature (+,-).

Is that a fair understanding of the word "split" ?

Split-octonians can't be associated with any Clifford algebra, because they aren't associative. But they're the next step in applying the Cayley-Dickson construction which gives the chain split-complex -> split-quaternion -> split-octonian. So that is the interpretation of "split" there. Is this correct, or are there further implications and connections that I'm not yet seeing? Jheald 08:45, 20 May 2007 (UTC)

According to Split-complex number the term "split" comes from the "signature" of the number system; however, no reference is provided, and it is also not clear which definition signature is meant. It appears to me that the definition of signature using 2-forms gives reason to the term "split", where complex numbers would have signature (+,+), split-complex numbers would have (+,-), quaternions (+,+,+,+), split-quaternions (+,+,-,-), and so forth (applying the Cayley-Dickson construction). The lower dimensional constructs are e.g. contained in Clifford algebras, however, the classifying construction is Cayley-Dickson on split-complex numbers. We should be careful not to mention Clifford algebras too early, thereby overstating their relation. Clifford algebras are important, and there is an overlap in two constructs (split-complex and split-quaternions), but that's where it ends. Thanks, Jens Koeplinger 15:04, 20 May 2007 (UTC)

IMHO the move from "Coquaternion" to "Split-quaternion" is not entirely unfounded, since interest in split-complex algebras has grown considerably in physics in recent years (e.g. click here for a search on "split octonion" at http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/all:+AND+octonion+split/0/1/0/all/0/1 arxiv.org]). This could underwrite notability of the newer term. Nevertheless, my vote for this article would still go towards keeping it at "Coquaternion", due to naming by the discoverer), and mention the identity with split-quaternions in the article. I've just not seen the new term "split-quaternion" used enough on its own (I've seen "split-octonions" and in general "split-complex" algebras more frequently). Thanks, Jens Koeplinger 01:31, 20 May 2007 (UTC)

Surely another apology is due from me for dropping the term "split form" into the conversation without providing links or explanations, and promptly promising to bow out! I've been kind of busy...

First split signature, for inner products (by which I mean nondegenerate symmetric bilinear forms), refers to inner products with signature (p,p), (p,p+1) or (p+1,p), i.e., equal or nearly equal numbers of pluses and minuses. As Jens points out, the split algebras have natural (i.e., automorphism-invariant) inner products of signature (1,1), (2,2) and (4,4), just as the "pure" algebras have inner products of signature (2,0), (4,0) and (8,0).

Second the associated Lie groups O(p,p) and O(p,p+1), and their Lie algebras, turn out to be the so called split forms. The articles on Lie theory seem to be rather undeveloped so far, but the article Real_form_of_a_complex_Lie_algebra at least redirects to a useful list. The "split form" is the real form whose real rank is equal to its complex rank. Roughly speaking, this means that the elements of a Cartan subalgebra have real eigenvalues, i.e., their characteristic polynomials split over the reals (hence the name). For example, the split form of sl(n,C) is sl(n,R).

The automorphism groups of the split quaternions and the split octonians are subgroups of O(2,2) and O(4,4), and are themselves split forms, namely SO(1,2), and the split form of G_2.

Finally, as noted by Jheald, the association with Clifford algebras (when it exists) also involves Clifford algebras associated to split signature inner products. This is related to the appearance of SO(1,2) above.

The serendipitous consonance of language between split signature and split form is one of the reasons the terminology has become popular, indeed, I claim, dominant.

PS. I support the merge with para-quaternion. Geometry guy 16:22, 20 May 2007 (UTC)

Thanks a lot for providing more info on where the naming "split" comes from! As it looks, we're today having the terms "split-quaternion" and "para-quaternion" used in different research contexts, yet describing an isomorphic algebra. Certainly nothing anyone would have apologize for ... it looks like a decision on which is the "better" term is doomed already. Thanks, Jens Koeplinger 01:14, 21 May 2007 (UTC)

para- is used to some extent in geometry (paraconformal or paraquaternionic manifolds), but I don't think it is used so much in algebra.

Anyway, in case it is not clear, I would be in favour of para- and co- redirecting to split-. At some stage, it might be nice to have a page analogous to normed division algebra which lists all the split algebras, and their properties. The Cayley-Dickson process also discusses the split case. I find the "split" terminology more descriptive and hence more neutral, but I'll be curious to see whether I have provided enough mathematical justification to satisfy Rgdboer's concerns.

Thanks again for the additional info. I've looked around a bit at topics where there are - in the widest sense - many individual "components" that are part in several different "programs" (like split-quaternion/split-complex extension, para-quaternion/paraquaternionic manifold, etc). I've come across chemicals, like e.g. Hydrogen, where its key properties are contained in a template insert on the right. Maybe that's what we need for our number systems? We cuold e.g. make one template for co-/split-/para-quaternion, list all its other classifications (including Clifford, Lie), its algebraic properties. Then, we could actually leave the individual pages for co/split/paraquaternion, list a bit of history and how it came that it has its different names etc. You pointed out yet another place at which the split-complex contruction is contained, so we might be able to remove the repeated explanation of it away from split-complex number, split-octonion, and Cayley-Dickson construction and instead have one article that is referenced from the template insert. ... Along these lines ... Obviously, some more planning is needed, since many pages are affected. ... We do want to make sure that all the different names for the same thing are represented reasonably, in particular how it came like it; but at the same time offer a one-click link to the related program (whether Cliffor, split-, para-, or whatever other).

This is good stuff, I'm glad I won't be getting bored over the summer :) - - - Thanks, Jens Koeplinger 12:50, 21 May 2007 (UTC)

[edit] Inner products

My thanks to G-guy for his clarification about the inner products/norms. One question, though. I know that for Clifford algebras, I can compute the signature ||s||² for each base s of the algebra, by starting with the known quadratic forms for vector bases, ||p||² = +1 or -1, and then using the fact that under Clifford multiplication of elements these signatures also multiply, ||pq||² = ||p||²||q||², revealing the signatures for bivector and higher bases.

But when one of these algebras cannot be related to a Clifford algebra (or half a Clifford algebra), where then do the signatures for its bases come from. Are they then implicit in the definition of the algebra? Or are they a further choice still to be settled, after the composition rules of the algebra have been fixed? -- Cheers, James Jheald 23:29, 22 May 2007 (UTC).

For (split-)octonions the quadratic form is still multiplicative; Cayley-Dickson constructs beyond that, however, don't have a multiplicative modulus at all, so I've got to pass your question on. I've never seen a thing like split-sedenions. Has anyone seen investigation into split-(2^N)-ions with N>3? Thanks, Jens Koeplinger 01:12, 23 May 2007 (UTC)

Second question, what is the right word for ||s||² ? For s a general member of a Clifford algebra, I think it equals where s~ is the reverse of s, and <>₀ means scalar part of. This can also be written where is the contractive inner product.

I'm wary about calling it a norm or a modulus, because no square root is taken, and because ||s||² could be negative. It's not simply a run-of-the-mill inner product, because of the need to include the reversal operation. Quadratic form might be right, but does this not more naturally suggest just s², without the reversal, and/or including non-scalar terms? Jheald 08:49, 23 May 2007 (UTC)

Either square-norm or quadratic form will do: the problem with the former is that it seems to suggest nonnegativity (but then so does the notation ||s||²); the problem with the latter is that it does not specify nondegeneracy. The notation is just a clean way of writing this square norm: expanded with respect to a suitable basis for the Clifford algebra, it will be a sum of s_i^2 terms (with signs). The reversal is a red herring as well: this is needed in the positive definite case, when is an inner product on the Clifford algebra. Geometry guy 13:26, 23 May 2007 (UTC)

Hmmm. I think you're going a bit fast, saying the reversal is just a red herring. The two most usual inner products, at least for real Clifford algebras, are sums over the contributions for each pair of grades k and l of the general multivectors A and B, the contributions being as follows:

• for the contractive product (the "computer scientist's inner product"),

  (zero if l < k),

• for the dot product (the "physicist's inner product"),

Neither contains reversal; but reversal is nevertheless required to get the right signs on the contribution to the square norm from bivectors and higher grade bases: if A=e₁e₂ then . But . So the square-norm is rather a special product.

Secondly, I'm also wondering if I may have been too hasty in asserting that ||PQ||² = ||P||²||Q||² for general multivectors P and Q. ||PQ||² = ; whereas ||P||²||Q||² = . And in general . The two will be equal if Q is a blade -- ie if it can be factorised as a pure Clifford product of vectors. But if Q is not a blade, then may contain cross terms which are not of grade 0. However, it seems very likely that application of an appropriate P and P~ may rotate these extra cross terms into resulting terms which are of grade 0. Such contributions would break the equality ||PQ||² = . Does this mean I've missed something? Or is that equality, which I have jumped to, not necessarily a generally correct assertion? Jheald 00:28, 24 May 2007 (UTC)

Thanks for pointing these out, but I would call these interior products rather than inner products, since they are not scalar-valued, although one could take the scalar part to obtain an inner product. This would differ from a definition using reversion by switching some signs according to the grade mod 4. One could also use conjugation instead of reversion to get other signs. I agree that it is most natural to use reversion.

For the second question, you are right that the equality does not always hold, cf. Hurwitz's theorem for composition algebras. Geometry guy 14:46, 28 May 2007 (UTC)