Talk:Hyperbolic quaternion

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

The answer to the question is within the article: "For example, (ij)j = kj = − i, while i(jj) = i."
 * 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


 * $$q(q^*)=a^2-b^2-c^2-d^2$$

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)

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).

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)

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)

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)

Algebra cuts
Going over the article I found that the idempotents identified did not satisfy the idempotent property. A coefficient of 1/2 was missing. Rather than fill it in, I re-evaluated the algebraic notes and now find them sufficiently covered by the facts of split-complex arithmetic (of which this structure is a naive extension). The thrust of this article should focus on its place behind Minkowski space and as a chapter in mathematical physics that serves as good reference in the development of matrix and tensor methods. The subjects of operational closures pale in significance compared to the milestone in concrete spacetime structures that this object represents.Rgdboer (talk) 22:51, 20 January 2008 (UTC)

Hyperbolic split-quaternions?
The hyperbolic quaternion multiplication table looks exactly like that of standard quaternions, but with all the signs on the main diagonal (save the top-left corner) swapped. Would it be possible to do this to the split-quaternions as well, giving the multiplication table to the right? Since there's been over a century and a half since quaternions were discovered, and this algebra doesn't seem to appear in the historical texts, I assume it is not particularly useful or interesting. For example, this algebra is not alternative, since (ij)j = kj = i while i(jj) = −i (the same example as the hyperbolic quaternions). I do not, of course, intend to talk about this algebra in the article, since nobody apparently ever cared about it; I am merely trying to confirm my current understanding of the topic. Double sharp (talk) 15:11, 31 March 2016 (UTC)
 * P.S. The exact same example in Rgdboer's post under "Modulus not multiplicative" shows that that problem also afflicts the "hyperbolic split-quaternions" I described, which would certainly explain why this case, though new, appears to have been overlooked as offering no interest (which I find myself, admittedly very far from being an expert, inclined to agree with). Double sharp (talk) 15:21, 31 March 2016 (UTC)
 * Yes, you seem to have found something new. As for the notability of the algebra discussed in this article, the events recounted in § Historical review serve up a chapter in history of mathematical physics. — Rgdboer (talk) 22:13, 31 March 2016 (UTC)

Blackboard bold symbol
The French Wikipedia template uses a blackboard bold M to stand for the hyperbolic quaternions. This makes some sense (H for the quaternions honours Hamilton, and so this M is presumably honouring Macfarlane), but I have some doubts if this is really a common symbol, especially since the hyperbolic quaternions do not appear to be a very-often-investigated algebra. Similarly, I have some doubts about the template's use of B for the biquaternions, and the use of a strikethrough to denote a split system. Double sharp (talk) 15:33, 31 March 2016 (UTC)
 * In this context M frequently refers to Minkowski, but details given here justify reference to Macfarlane. As for standard symbols, beyond ℕ, ℤ, ℚ, ℝ, ℂ, and ℍ, there are the matrix rings M(n,F) of n x n matrices over a field F. You might look over glossary of algebraic geometry also. — Rgdboer (talk) 22:22, 31 March 2016 (UTC)

How is this algebra important or natural?
As far as I can tell, this algebra is only of interest to historians. I don't see why it's so important that it has to feature in the "Number systems" box. It's perhaps notable enough to merit an article, given that a few papers were written about it at the turn of the 20th century, but not because of any intrinsically mathematical features it has.

From a mathematical viewpoint, this algebra is quite ill-behaved: It lacks associativity, commutativity, or a quadratic norm. Why would anyone need this? — Preceding unsigned comment added by 62.172.100.253 (talk) 10:21 16 May 2019 (UTC)


 * Hello London: the algebra does have a quadratic norm $$q(q^*)=a^2-b^2-c^2-d^2$$ which happens to be the same as in Minkowski space. The algebra also has hyperbolic versors, which accomplish the same planar mapping as a Lorentz boost. As for the Number systems template, the row of hypercomplex numbers contains algebras that stimulated the developments of linear algebra, group and ring theory, Lie theory and relativity. Evidently you found this algebra to be "un-natural" ! — Rgdboer (talk) 21:13, 16 May 2019 (UTC)

It seems mainly of historical interest; I've never seen any modern work using it. But I'm very glad this article exists, because this algebra played an interesting role in the developments connecting algebra to physics. I'd say it's an important example of a wrong turn. John Baez (talk) 06:13, 21 October 2019 (UTC)

Useless multiplication table
The table titled "Hyperbolic quaternion multiplication" is useless, because it neglects to specify whether the row label or the column label comes first in the product of two quaternions whose value is listed.

I hope that someone familiar with this subject can fix this omission. — Preceding unsigned comment added by 2601:200:c082:2ea0:e9dd:6b98:c1ee:9d73 (talk) 21:26, 16 September 2023 (UTC)


 * First, the convention is row times column. But second, the same content is explicitly repeated in the following section. –jacobolus (t) 01:57, 17 September 2023 (UTC)