Talk:Sedenion

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Could you describe what are they used for ? --Taw

Also, what is the etymology of the name? Incidentally, I'm [RobertAtFM|http://wiki.fastmail.fm/wiki/index.php/RobertAtFm] on the [FastMail.FM Wiki|http://wiki.fastmail.fm/].


Yes, I agree: this page needs more information. I'd love to learn more about these things, but this page is barely more than a stub. --AlexChurchill 10:46, Jul 27, 2004 (UTC)

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[edit] multiplicative inverses and zero divisors

I wonder about the following feature of the sedenions as claimed in the article: They have multiplicative inverses and at the same time zero divisors.

In a matrix algebra, these two features cannot coexist, since a divisor of zero necessarily has determinant zero and thus it is not invertible.

I"d like to see an example of two sedenions with inverses and a product of zero. --J"org Knappen

What the article says is:

The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors.

Take a look at zero divisors which demonstrates the principle with matrices. A given sedenion might be a zero divisor when paired with one particular other, but nothing stops either having an inverse as far as I can see. HTH --Phil | Talk 11:56, Jan 11, 2005 (UTC)
Seen that -- nice demostration of what is possible in infinitely many dimensions. My argument above is for finite matrices (what"s the determinant of an infinite matrix, anyway?) I just like to see an example for sedenions, too --J"org Knappen
J"org - Matrices form an associative algebra. Sedenions are non-associative. You are not able to map a sedenion algebra on matrices (as opposed to e.g. octonions or quaternions). For examples of calculations with sedenions see e.g. the two references to articles from K. Carmody. Please note, however, that the sedenions in there are of a different type as the ones displayed in the multiplication table (I'll try to have this corrected). --65.185.222.50 13:19, 12 September 2005 (UTC)

Corrigendum

The multiplication table given in the article applies to sedenions of the type discussed by Imaeda/Imaeda, however, the sedenions discussed by Carmody are of a different type, first proposed by C. Musès. The following reference should be added to establish the correct originatorship of the latter sedenion type: C. Musès, Appl. Math. and Computation, 4, pp 45-66 (1978) --65.185.222.50 13:19, 12 September 2005 (UTC)

[edit] history

when were they discovered? are there larger algebras with the same properties? 83.79.181.211 19:05, 29 September 2005 (UTC)

About the term "sedenion" and the discovery; as far as I know, the cited articles are the earliest publications I could find that go deeper into arithmetic laws of the sedenions, and thereby solidify the term (so it's quite recent). Surely, as part of the Cayley-Dickson construction, their existence was discovered earlier within that program. Any earlier known uses of the term "sedenion"? Thanks, Jens Koeplinger 12:40, 15 September 2006 (UTC)

There are larger algebras with the same properties, in fact an infinite number of them. One can perform the Cayley-Dickson construction on sedenions to get a 32-element algebra, and again to get a 64-element algebra - in fact, one can get an algebra of 2n for any non-negative n. n=0 gives the reals, n=1 gives complex numbers, n=2 quaternions, n=3 octonions, n=4 sedenions and so forth. --Frank Lofaro Jr. 22:33, 2 March 2006 (UTC)

[edit] Duo-tricenians??

This article says:

"Like (Cayley-Dickson) octonions, multiplication of Cayley-Dickson sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of being power-associative." In turn, is the property of being power-associative an operation that the duo-tricenians no longer have?? Georgia guy 19:31, 11 September 2006 (UTC)

Hi. All Cayley-Dickson construction products remain power associative, including the 32-ions. This does - of course - not apply to modified constructs, like e.g. the split-octonions, which contain nilpotents. Thanks, Jens Koeplinger 12:37, 15 September 2006 (UTC)


[edit] Power-associative conic sedenions ?

The article claims that conic sedenions are alternative and flexible. It also claims them to not be power-associative. I assume the conic sedenions are an algebra i.e. multiplication distributes over addition and commutes with scalar multiplication. Doesn't general di-associativity (and hence power-associativity) follow from any two of the alternative laws (left, right, and flexible) in an algebra, giving an alternative algebra? 85.224.17.70 (talk) 19:52, 19 January 2008 (UTC)

Hello - maybe there is a terminology conflict here? I've taken the term "power associative" from the Imaeda/Imaeda publication (personal web space: http://www.geocities.com/zerodivisor/sbasicalgebra.html , formally published: http://dx.doi.org/10.1016/S0096-3003(99)00140-X ), but the definition that is given here is different: In the sense that x(x(xx)) = (xx)(xx) you're absolutely correct, of course. The relation used by Imaeda, SnSm = Sn + m, however does not hold anymore in general for conic sedenions (or complex octonions), due to the presence of nilpotents (e.g. exp((\pi / 2)(i_1 + \varepsilon_2)) = 1 + (\pi / 2)(i_1 + \varepsilon_2) \ne exp((\pi / 2) i_1) * exp((\pi / 2) \varepsilon_2). The stronger SnSm = Sn + m implies x(x(xx)) = (xx)(xx), so I'm not sure what the correct terminology would be ... help appreciated! Thanks, Jens Koeplinger (talk) 20:50, 19 January 2008 (UTC)

[edit] Baez's quote

If octonions are the crazy uncle that no one lets out of the attic, would sedenions be the serial killer maximum-security prison escapee that no one even lets in the house? Phoenix1304 (talk) 16:11, 9 April 2008 (UTC)

Ha! That seems to go in the right direction. Because they contain zero divisors, they're certainly "convicted" of some sort of violation ... :) --- feel free to post any concerns or more detailed questions you may have. Thanks, Jens Koeplinger (talk) 02:53, 10 April 2008 (UTC)
I'm actually less confused about these than I was about the other thngs. Sedenions seem to be "closer" to octonions than octonions are to quaternions, so it's not a huge leap. Phoenix1304 (talk) 15:37, 13 April 2008 (UTC)