Talk:Hyperbolic quaternion

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[edit] Cleanup

This article is built on what seems like a faulty premise that I'm having great trouble understanding.

  • Claims of non-associativity, but I don't see what's non-associative, there are no examples given. Ohh ... I see ... its non-associative in the sense that a Lie algebra is non-associative ... right. But its still a potentially confusing statement that needs to be made more explicit up front.
  • Odd notation, such as ij = ( − j)i, but surely ( − j)i = − (ji) ?? It is claimed that "the the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4." and since -1 is a real number, I presume that -1 times j is equal to -j. So why the confusing (-j)i notation when -ji seems to suffice ?
  • Talk of a four-dimensional basis set {1,i,j,k} which is somehow not enough and so now there's an eight-dimensional basis set {1,i,j,k, -1, -i, -j, -k}. Again, I can sort-of understand the point here, but its confusingly made.

The relationships resemble the Lie algebra sl(2,C), since the article claims

q(q * ) = a2 − b2 − c2 − d2

which us special-relativity folks interpret as the well known

sl(2,C) \otimes sl^*(2,C) = so(3,1)

product that basically says the product of two spinors is a vector. However, sl(2,C) is a complex lie algebra, whereas the hyperbolic quaternions are real. So this is quite confusing. All this should be cleaned up and clarified. linas 14:07, 17 August 2006 (UTC)

Linas, thanks very much for initiating this call for clean-up. I believe the numbers work, but also agree that they need clean-up. The current confusion appears to arise that each section refers to a different publication about this system, and uses the terminology therein; which is - as so often - not consistent within all publications. But that's no problem, that's our job to figure out, I guess :). Let me start a separate section here in the talk page about algebraic properties, so we can leave this thread for more general remarks. Thanks, Jens Koeplinger 16:50, 17 August 2006 (UTC)
Actually, I don't see any further inconsistencies at this point. There's Knott's theorem, but that's clearly about something else, so there is no inconsistency there. I am now satisfied with the article, and I state below, I'm done. linas 01:33, 18 August 2006 (UTC)
It's a great improvement, thanks for the detailed work. If you don't mind, I'll later make some wording in the introduction more neutral, and add the remarkable property that these hyperbolic quaternions are the only known quaternionic system with multiplicative modulus where two non-real roots of +1 multiply to another non-real root of +1. Actually, right now I'm also not aware of any 8- or 16-dim system that would do this, either, but I won't say anything about this until I've looked at it more deeply sometime later. Thank you very much for your help. Jens Koeplinger 01:44, 18 August 2006 (UTC)

[edit] Algebraic properties

This section lists algebraic properties of A. MacFarlane's hyperbolic quaternions. Please correct anything that is wrong here, but I suggest to add general comments outside this section, so it can later easier be merged into the article.

Hyperbolic quaternions after A. MacFarlane are a four dimensional distributive, non-ccommutative, and non-associative vector space over the reals, with one real base 1 and three non-real bases i, j, k.

The multiplication table is:

ii = 1 , ij = k , ik = -j

ji = -k , jj = 1 , jk = i

ki = j , kj = -i , kk = 1

All non-real bases are anti-commutative (ij = -ji = k, jk = -kj = i, ik = -ki = -j), therefore multiplication is generally non-commutative.

The non-real bases are anti-alternative, e.g.: (ij)j = kj = -i but i(jj) = i - therefore multiplication is generally not alternative and subsequently not associative.

The modulus | | z | | of a number z with coefficients (a,b,c,d) to bases {1,i,j,k} is defined as

||z|| := \sqrt{a^2 - b^2 - c^2 -d^2}

and is multiplicative, i.e. for any two hyperbolic quaternions x and y the product of the respective moduli is the modulus of the product:

||xy|| = ||x||~||y||

The algebra contains zero divisors

(1 + i)(1 - i)~= 0

and idempotents

(1 \pm i)^2 = (1 \pm i).

The algebra is closed under addition, subtraction, and multiplication. With the exception of zero, its zero divisors, and its idempotents, the algebra is also closed under division. In contrast to complex numbers or quaternions, the algebra is not closed under exponentiation, since e.g. irrational exponents of -i, or roots with even denominator, are undefined within the system (similar to the reals).

[edit] Toward clarity

Thank you both for your care. I have removed the unnecessary parentheses. Also removed contradiction tag since it is unspecified in comments. Jens, thanks for the non-associative example and algebraic properties that do hold. Keep in mind that this article traces a development that preceeded all our modern insight on what is desirable in a structure. Joining up split-complex arithmetic with quaternions in this fashion wears thin soon with the non-associativity, but for a while it looked very promising. The split-complex arithmetic at that time was supressed due to fear of zero-division, and linear representation was not at all common. So hyperbolic quaternions were a practice exercise before biquaternions became the ring of choice for a while. Once tensor algebra took over all this was put away. Rgdboer 22:55, 17 August 2006 (UTC)

Thank you for sharing your historical knowledge! I was just updating the article and learned the use of the { {inuse} } tag - the hard way ;-) But I thank Linas for starting, of course. I disagree with taking some wording directly to the article, like "all modern insight" etc; I'll propose changes there later. To my knowledge, they are the only quaternionic system where two non-real roots of 1 multiply to another non-real root of one, but the number system yet offers a multiplicative modulus. I believe this to be a very special property; even if they were to turn-out to be a dead end. Compare them e.g. to coquaternions, split-octonions, or conic sedenions which also have a multiplicative modulus, yet any two non-real roots of 1 multiply to an imaginary base (a root of -1). Thanks, Jens Koeplinger 01:19, 18 August 2006 (UTC)

[edit] Thank you

Thanks. I cut-n-pasted much of the above into the article, shuffled some text around, made a clearer distinction between the algebraic review and the historical commentary, and some general copyedits. This is now in a form that is entirely comprehensible, at least to me, so I removed thecleanup/confusing tag. linas 01:21, 18 August 2006 (UTC)

[edit] Modulus not multiplicative

Consider (1 + i)(i + j) = i + j + 1 + k . But the modulus of 1 + i is zero. On the right hand side the modulus-square is 1 - 3 = -2. The asserted identity does not hold. In the linear rings of matrices the modulus-square arises as the determinant. In those cases, not here, the multiplicative property of the determinant corresponds to a multiplicative modulus.Rgdboer 23:10, 30 August 2006 (UTC) Since the product q (q*) gives the essential quadratic form, which may be negative, and produce imaginary values upon being square-rooted, the introduction of a "modulus" is unnecessary and potentially confusing. Failing the previously asserted property, there is no motivation to introduce it at all, so I have deleted it.Rgdboer 23:20, 30 August 2006 (UTC)

Robert - thanks for finding this error and correcting it. I agree with everything you said and did. Now I understand why interest in these numbers was lost after a while, as you detailed in the introduction of the article. I'll check and make sure this error is not stated anywhere else here. Thanks, Jens Koeplinger 23:51, 22 October 2006 (UTC)
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