Talk:Cayley–Dickson construction

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Inventor?

Cayley is surely Arthur Cayley, 1821-1895, Dickson must be Leonard Dickson, 1874-1954.

But who invented it, and why is it called Cayley-Dickson? — Preceding unsigned comment added by 63.249.19.189 (talk) 11:07, 28 January 2003

Multiplication formula

In the (german, transl. from russian) book of Kantor and Solodownikow, I found the following formula for the multiplication of pairs: $(a,b)(c,d)=(ac-d^{*}b,da+bc^{*})$ which is different from the formula given here. How to decide which one describes the Cayley-Dickson construction correctly? --J"org Knappen

Found it out: The two formulæ are æquivalent -- JKn.

The argument for equivalence was wrong, thus the problem still remains -- JKn

They generate isomorphic structures (identical up to signs of bases). Taking i, k, l as (0, 1), (0, 1), (0, 1) resp. and the others by analogy, the K-S formula generates the standard basis (ij=k, etc.) on

\mathbb {H}

(the Wikipedia formula gives ij=-k) but nonstandard on

\mathbb {O}

(il=(0,i) where the Wikipedia formula gives il=(0,-i)=-li). I rattled off a quick Python script to generate the sedenion multiplication tables for the two formulas and it turns out the table in Sedenion is that generated by the K-S formula. The two tables are identical except that the one has opposite sign to the other off the leading diagonal, row and column; equivalently, replacing all the non-1 units with their negatives.

The inductive proof is fairly easy (RTP:

x*_{W}y=-x*_{KS}y,\pm 1\neq (x=(a,b))\neq (y=(c,d))\neq \pm 1

. Assumption: this holds for the previous level (i.e. for x=a, y=c, etc. Method: each of a, b, c, d can be 0, 1 or another complex unit from the previous level, etc.,)

Of course, this raises the question: which formulas give the correct algebraic structure? The main conditions are that norms work, that 1 works and that addition distributes over multiplication. These are satisfied by keeping the shape of the formula, but applying three (independent, Abelian) transformations: moving the conjugate operator on the db term (d*b/db*), swapping the order of the terms in the second part (da+bc*/ad+c*b), and moving the conjugate operator between the terms in the second part (da+bc*/da*+bc). This gives 8 isomorphic constructions; the same inductive proof demonstrates that the resulting structures are again equivalent up to replacing some bases with their negatives. Yet more isomorphic structures are available, however, by using different formulæ for the construction at different levels. Not that many, though; only 8^n as opposed to 2^(2^n-1) possibilities flipping bases. Which 8^n, and are all different? Uh... wish I could answer that. Not my field. Fun, though. -- EdC 02:26, 25 April 2006 (UTC)[reply]

The octonions have two natural bases, $\{1,i,j,k,l,il,jl,kl\}$ , and $\{1,i,j,k,l,li,lj,lk\}$ , each of which has a different multiplication table, and therefore needs a different formula for the C-D construction. It becomes more clear if you explicitly interpret the pairs as addition: the formula given in the article means

(a+lb)(c+ld)=(ac-db^{*})+l(a^{*}d+cb),

whereas the K-S formula means

(a+bl)(c+dl)=(ac-d^{*}b)+(da+bc^{*})l.

Thus both variants describe the same object. -- EJ (talk) 15:39, 25 February 2008 (UTC)[reply]

A naive derivation (based on a matrix, or operator, approach) gives:

(ac-bd^{*},c^{*}b+da)

. If

F

is commutative this coincides with the one in the article, but if

F

is not commutative then this becomes problematic.

The rational is based on starting the construction by creating an isomorphism between

F^{2}

and

F^{2}\times F^{2}

as follows

(a,b)\leftrightarrow {\begin{bmatrix}a&-b\\b^{*}&a^{*}\end{bmatrix}}

and using usual row-column matrix-matrix multiplication and the conjugation rule

(ab)^{*}=b^{*}a^{*}

to get the result

{\begin{bmatrix}a&-b\\b^{*}&a^{*}\end{bmatrix}}{\begin{bmatrix}c&-d\\d^{*}&c^{*}\end{bmatrix}}={\begin{bmatrix}ac-bd^{*}&\dotsb \\(c^{*}b+da)^{*}&\dotsb \end{bmatrix}}

77.102.36.170 (talk) 12:35, 6 June 2024 (UTC)[reply]

Construction

It isn't clear to me what the notion of construction in this (algebraic) context means. I went to the disambiguation page for construction to find out, but it only lists the geometric sense in which I'm already familiar. Can someone provide some background pointers in this article? 70.250.189.176 (talk) 05:05, 21 March 2010 (UTC)[reply]

construction is just used in its common English sense, i.e. it is something that is constructed or made. So the complex numbers are made from the reals, the quaternions are constructed from the complex numbers, etc.. The article describes how this is done in these cases, for the octonions and the general case.--JohnBlackburne^words_deeds 10:28, 21 March 2010 (UTC)[reply]

Unfortunately, that explanation doesn't really fly, for two reasons that I'm aware of:

it doesn't distinguish between allowable and unallowable constructions
the ordinary sense of the term construction doesn't allow for the "loss" of properties, as is seen here--that is, the Cayley-Dickson construction seems like more of a decomposition than a construction. 70.250.238.188 (talk) 15:43, 21 March 2010 (UTC)[reply]

I think you're trying to read too much into it. The word construction to me just means that it is a method for making e.g. complex numbers out of reals. There are many other constructions "allowed", but this one generates a series of algebras with interesting properties so is notable. As for losing properties, when you make complex numbers out of reals the reals do not change, so do not lose properties. You make the complex numbers which have their own properties.--JohnBlackburne^words_deeds 15:56, 21 March 2010 (UTC)[reply]

Perhaps I am trying to read too much into it, but I think it proves my point, that it isn't clear. 70.250.238.188 (talk) 17:24, 21 March 2010 (UTC)[reply]

It is an encyclopaedia article not a text book chapter, so is far briefer than it could be. If it isn't clear then that's understandable - there are many articles I find not clear as my expertise in their subject area is lacking, and they are too brief to gain an understanding of the subject from. You could follow up the references, or read other related articles - octonion and sedenion cover the particular algebras usually derived from the construction. If you have questions you can ask here or at the reference desk - the reference desk in particular is visited by far more people able to answer mathematical questions at all levels.--JohnBlackburne^words_deeds 19:35, 21 March 2010 (UTC)[reply]

OK, that's just rude. Why are you getting so defensive? The article certainly won't get any better if people don't ask questions, right? I'm not an expert, yes. Who do you think your audience is? Experts? If you re-read my comments, you'll notice that I did consult related articles, and none of them has done a good job of explaining, which is why I asked the question in the first place. I just think it's kind of backwards to define a finite group from the starting point of an infinite one, and I'd like to find a substantive reason to think I'm wrong (and learn something new). Unfortunately, this article doesn't help. 70.250.238.188 (talk) 09:59, 22 March 2010 (UTC)[reply]

Not sure why you think that the C-D construction ever defines a finite group. The complex numbers, quaternions, octonions etc. are all infinfite sets. Each algebra has a finite dimension when considered as a vector space over the real numbers (so you need two real numbers co-ordinates to specify a complex number; four to specify a quaternion; eight to specify an octonion etc.) but they all contain an infinite number of members. Each algebra in the sequence contains each of the previous algebras as a sub-algebra. JohnBlackburne is correct - "construction" just means "a method of making"; in this case, it is a method of making a sequence of algebras with certain properties. Gandalf61 (talk) 10:38, 22 March 2010 (UTC)[reply]

I'm not sure why I thought that, but I see now that you both are correct. I'm guessing that it has to do with the "numeral" prefixes distorting my rationality. 70.250.199.181 (talk) 01:40, 23 March 2010 (UTC)[reply]

I'm sorry if my comments seemed too terse, but as regards your original query I think we've covered the use of "construction" pretty well. Although all articles should be as accessible as possible this is a fairly obscure topic which mostly arises in the construction of the octonions: It's overkill and a non-obvious way to explain complex numbers and quaternions. It only has these few examples of its use (no-one really cares about the sedenions), all of which are given in the article, so there's not much to add. The way it is presented in these few examples is very straightforward, using basic algebraic properties, so could not be much simplified.--JohnBlackburne^words_deeds 11:21, 22 March 2010 (UTC)[reply]

I understand well enough to agree that the algebraic description here is pretty good. I'm still not sold on your explanation of construction, but it makes enough sense that I'll leave it alone. I'm not sure why you feel this construction is overkill, though; I've rarely seen a mathematical observation to be meaningless. That's why I appreciate the work that's gone into this article. Thanks. 70.250.199.181 (talk) 01:40, 23 March 2010 (UTC)[reply]

Mapping Cayley-Dickson representation of quaternions to 1,i,j,k representation?

I understood the mapping for the complex numbers, (a, b), where a and b are real, corresponds to a complex number a + bi (where a and b are real). However it's not clear to me how (a, b), where a and b are complex, corresponds to a quaternions p + qi + rj + sk (where p, q, r, s are real).

I'm also curious about the inverse map. Is there a canonical or obvious way to represent the 1, i, j, k quaternion bases in terms of pairs of complex numbers? By trial and error, I found that 1 <-> (1, 0) i <-> (i, 0) j <-> (0, 1) k <-> (0, i) seems to satisfy i² = j² = k² = -1 and ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j

Should some discussion about mapping and inverse mapping be added to the article?

Dgrinstein (talk) 04:47, 5 May 2010 (UTC)[reply]

The most convenient way is to interpret (a + bi, c + di) as a + bi + cj + dk, which is equivalent to your assignments for i, j and k. (The assignment of 1 to (1,0) is forced.) The article probably ought to mention this. --Zundark (talk) 08:40, 5 May 2010 (UTC)[reply]

The 1=(1,0) is not forced if one equates a real number x with the sequence x,0,0,0, . . . and equates the ordered pair (x,y) of two sequences x₀,x₁,x₂, . . . and y₀,y₁,y₂, . . . with the shuffled sequence x₀,y₀,x₁, y₁, . . . . In this context, 1=(1,0) and one has, for the basis vectors e₀=1, e_2k=(e_k,0) and e_2k+1=(0,e_k). Johnwaylandbales (talk) 20:32, 10 January 2016 (UTC)[reply]

Split Cayley-Dickson construction

It seems to me (after thinking about it and trying it) that if you replace the formula $(a,b)(c,d)=(ac-d^{*}b,da+bc^{*})\,$ with $(a,b)(c,d)=(ac+d^{*}b,da+bc^{*})\,$ , and apply it repeatedly to the reals, you get the split composition algebras (split-complex numbers, split-quaternions, and split-octonions). If this is true, does it appear in any reliable source, as it would tie nicely together how the Cayley-Dickson construction easily zeroes in on the best algebras (the ones that are actually new, instead of being reducible or matrix rings), and with a minor change zeroes in on the remaining ones which, though not the best, still have some nice mathematical properties. (When I try this on the split-complexes, I only get the split-quaternions: so it seems that swapping the sign anywhere along the path from R brings you to the split algebras, while you can only get to the division algebras if you never switch the sign.) Double sharp (talk) 15:15, 31 March 2016 (UTC)[reply]

Ah, I found a source for this. Double sharp (talk) 12:07, 1 April 2016 (UTC)[reply]

McCrimmon (2004) A Taste of Jordan Algebras looks like a fine reference with History and links to related areas ! — Rgdboer (talk) 20:42, 1 April 2016 (UTC)[reply]

Indeed it does, and it's even available online for free to boot. Double sharp (talk) 16:08, 2 April 2016 (UTC)[reply]

The section § 2.9 Forms Permitting Composition on pages 62 and 63 has wrong and misleading assertions. First, Hamilton biquaternions are isomorphic to M(2,C), not to the direct sum of H with itself (which is Clifford’s biquaternions, here split-biquaternions). Next, McCrimmon writes "A subsequent flood of (false!) higher dimensional algebras carried names such as quadrinions, quines, pluquaternions, nonions, tettarions, plutonians. Ireland especially seemed a factory for such counterfeit algebras." Compare this rant with our hypercomplex numbers article. On the other hand, the identification of complex numbers as binarions is spot-on and opens the door to broader use and understanding of the seven real composition algebras. — Rgdboer (talk) 20:40, 8 April 2016 (UTC)[reply]

Reference

The article cites Schafer (1995), yet R. Schafer only wrote 'An introduction to non associative algebras' (1966) to my knowledge. What reference is meant? — Preceding unsigned comment added by Primetimer (talk • contribs) 14:49, 13 May 2016 (UTC)[reply]

Evidently Dover Books republished the 1966 text in 1995:

Schafer, Richard D. (1995) [1966], An introduction to non-associative algebras, Dover Publications, ISBN 0-486-68813-5, Zbl 0145.25601

Twenty-nine years did nothing to diminish the book's value. — Rgdboer (talk) 01:38, 20 July 2016 (UTC)[reply]

The article also asserts that the General construction was done by Albert in 1942. See octonion algebra for references to Dickson 1927 and Zorn 1931. As this is a popular topic in abstract algebra, there may be editors with more references to contribute. — Rgdboer (talk) 01:30, 20 July 2016 (UTC)[reply]

Introduction paragraph (composition property)

The last sentence of the first paragraph seems to suggest that all algebras generated by the Cayley-Dickson construction are composition algebras. However, we learn further down that the sedenions are no longer a composition algebra. Hence, it might be better to change the sentence to something weaker such as "The first four algebras in this sequence are composition algebras frequently used in mathematical physics". I will leave it to someone more knowledgeable to implement this change. 141.5.38.53 (talk) 17:18, 11 July 2017 (UTC)[reply]

Thank you reader in Erlangen for noting this fact. Modification has been made. As this Project needs sharp eyes and succinct expression, you are encouraged to register an account and get a User page and Watchlist. — Rgdboer (talk) 23:03, 11 July 2017 (UTC)[reply]

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Question

Hi. I noticed the article state: The complex conjugate (a, b)* of (a, b) is given by

{\displaystyle (a,b)^{*}=(a^{*},-b)=(a,-b)} {\displaystyle (a,b)^{*}=(a^{*},-b)=(a,-b)} since a is a real number and its conjugate is just a.

This seems to be assuming familiarity with the complex numbers, which are being constructed here. Isn't this circular reasoning? I made an attempt at making a proof which used just the properties described here, with no reference to complex numbers, but I wasn't sure if it would be useful, and it would be original research probably... Thanks for reading. :) JonathanHopeThisIsUnique (talk) 22:37, 20 January 2018 (UTC)[reply]

Good point. Still, the construction proceeds in the general fashion that starts with an arbitrary field having the trivial involution for its "conjugation". See, for example, Complex plane#Quadratic spaces for illustration of the point. The section is establishing the binarion stage of the construction. The viewpoint was taken by Hamilton, and built up since then. — Rgdboer (talk) 01:58, 22 January 2018 (UTC)[reply]

Abraham Adrian Albert Generalization and the notion of signature

Reading the cited article of Albert(1942) I found that you are changing a sign in the definition. The correct definition published in the Albert paper is: $(a,b)(c,d)=(ac+\gamma d^{*}b,da+bc^{*})$ . This means that, if you chose $\gamma =i^{2}$ (where $i$ is the imaginary unit), then you get the Cayley-Dikson construction. On the other hand, if you chose $\gamma =1$ , you get a different algebra. This is clear for me in the article. The cited article of Schafer(1954) uses the same definition. Moreover, Albert explicitly states that $\gamma$ must be different from zero (this is a necessary condition for his Lema 1). On the contrary, you allows $\gamma$ to be zero. And this could be fine because, then, the Albert generalization also includes dual numbers with this choice. Even if I do not see all the consequences at this moment.

In conclusion, with the correct sign, one can understand the Albert generalization in the following way: let $\gamma _{i}$ be equal to one of {-1, 0, 1} for i=1,2,3...n (n any integer). This kind of sequences can be called a signature and can be represented also with the signs {-,0,+} only. Then, complex numbers have signature (-), quaternions have signature (-,-), octonions have signature (-,-,-); sedenions have signature (-,-,-,-), etc. Any Cayley-Dickson algebra has a signature with only negatives. On the other hand, split numbers have signature (+) and there can be also split numbers with split components (for which I do not have a name) with signature (+,+), etc. Moreover, dual numbers have signature (0) and there can be also dual numbers with dual numbers as components, this will have signature (0,0), etc. Finally, there can be also numbers with mixed signatures like (-,+) (which, I think, you call split-quaternions in the article) or numbers with signatures like (+,-,-,-).

As I understand things, Schafer shows that all algebras with signatures - or + (for any length and mixed in any order) are flexible in his 1954 article, and he was aware of this fact.

The inclusion of dual numbers in the Albert generalization could be more controversial. 17:35, 13 December 2018 (UTC)Crodrigue1 (talk)

What is Zorn(R)

In the section about modified Cayley-Dickson construction, it is said that the split octonions are isomorphic to Zorn(R). There is however no mention of it anywhere in the current version, nor any link. — Preceding unsigned comment added by 109.172.129.12 (talk) 15:47, 2 January 2019 (UTC)[reply]

Follow the link:Split-octonion#Zorn's vector-matrix algebra — Rgdboer (talk) 02:12, 3 January 2019 (UTC)[reply]

The link has been put in the article for easy reference. — Rgdboer (talk) 01:24, 4 January 2019 (UTC)[reply]

Synopsis

As it stands the article is long-winded like a university lecture and repetitive. The synopsis below is suggested as containing important missing information such as propositions with proofs:

The Cayley—Dickson construction is due to Leonard Dickson in 1919 showing how the Cayley numbers can be constructed as a two-dimensional algebra over quaternions. In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2ⁿ. For n = 2 it is an associative algebra called a quaternion algebra, and for n = 3 it is an alternative algebra called an octonion algebra. These instances n= 1,2 and 3 produce composition algebras as shown below.

The case n = 1 starts with elements (a.b) in F x F and defines the conjugate (a,b)* to be (a*, –b) where a* = a in case n = 1, and subsequently determined by the formula. The essence of the F-algebra lies in the definition of the product of two elements (a,b) and (c,d):

(a,b)\times (c,d)\ =\ (ac-d^{*}b,\ da+bc^{*}).

Proposition 1: For $z=(a,b)\ {\text{and}}\ w=(c,d),$ the conjugate of the product is $w^{*}z^{*}=(zw)^{*}.$

proof:

(c^{*},-d)(a^{*},-b)=(c^{*}a^{*}+b^{*}(-d),\ bc^{*}-da)\ =\ (zw)^{*}.

Proposition 2: If the F-algebra is associative and $N(z)=zz^{*},\ {\text{then}}\ N(zw)=N(z)N(w).$

proof:

N(zw)=(ac-d^{*}b,da+bc^{*})(c^{*}a^{*}-b*d,-da-bc^{*})\ =\ (aa^{*}+bb^{*})(cc^{*}+dd^{*})

+ terms that cancel by the associative property.

— Rgdboer (talk) 17:20, 4 August 2020 (UTC) Dickson 1919 — Rgdboer (talk) 18:30, 4 August 2020 (UTC) — Rgdboer (talk) 17:33, 27 August 2020 (UTC) — Rgdboer (talk) 17:30, 7 September 2020 (UTC)[reply]

Possible extension

The box on the top right should be extended to show the properties of algebras of this type of 32, 64, 128 etc. dimensions. I think that calculations have been made up to 256 dimensions. — Preceding unsigned comment added by 2A00:23C4:7C87:4F00:747C:5E75:53A5:B945 (talk) 13:25, 28 August 2020 (UTC)[reply]