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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
In the (german, transl. from russian) book of Kantor and Solodownikow, I found the following formula for the multiplication of pairs: which is different from the formula given here. How to decide which one describes the Cayley-Dickson construction correctly? --J"org Knappen
The octonions have two natural bases, , and , 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
whereas the K-S formula means
Thus both variants describe the same object. -- EJ ( talk) 15:39, 25 February 2008 (UTC)
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)
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)
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, e2k=(ek,0) and e2k+1=(0,ek). Johnwaylandbales ( talk) 20:32, 10 January 2016 (UTC)
It seems to me (after thinking about it and trying it) that if you replace the formula with , 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)
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)
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)
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)
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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)
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: . This means that, if you chose (where is the imaginary unit), then you get the Cayley-Dikson construction. On the other hand, if you chose , 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 must be different from zero (this is a necessary condition for his Lema 1). On the contrary, you allows 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 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)
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)
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):
Proposition 1: For the conjugate of the product is
Proposition 2: If the F-algebra is associative and
— 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)
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)
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Change 'octonians' in first paragraph of Synopsis to 'octonions' 210.23.160.153 ( talk) 05:01, 22 November 2022 (UTC)
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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
In the (german, transl. from russian) book of Kantor and Solodownikow, I found the following formula for the multiplication of pairs: which is different from the formula given here. How to decide which one describes the Cayley-Dickson construction correctly? --J"org Knappen
The octonions have two natural bases, , and , 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
whereas the K-S formula means
Thus both variants describe the same object. -- EJ ( talk) 15:39, 25 February 2008 (UTC)
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)
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)
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, e2k=(ek,0) and e2k+1=(0,ek). Johnwaylandbales ( talk) 20:32, 10 January 2016 (UTC)
It seems to me (after thinking about it and trying it) that if you replace the formula with , 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)
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)
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)
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)
Hello fellow Wikipedians,
I have just modified one external link on Cayley–Dickson construction. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
An editor has reviewed this edit and fixed any errors that were found.
Cheers.— InternetArchiveBot ( Report bug) 12:53, 1 August 2017 (UTC)
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)
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: . This means that, if you chose (where is the imaginary unit), then you get the Cayley-Dikson construction. On the other hand, if you chose , 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 must be different from zero (this is a necessary condition for his Lema 1). On the contrary, you allows 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 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)
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)
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):
Proposition 1: For the conjugate of the product is
Proposition 2: If the F-algebra is associative and
— 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)
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)
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edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
Change 'octonians' in first paragraph of Synopsis to 'octonions' 210.23.160.153 ( talk) 05:01, 22 November 2022 (UTC)