THE OCTONIONS. JOHN C. BAEZ. Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected. The expository article about octonions by John (Baez) that appeared in the AMS Bulletin (copy here, a web-site here) is one of the best pieces. After the octonions comes a dimensional number system called the ” sedenions”. They have some nice features, which I describe in this.
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Markus Rost, On the dimension of a composition algebra, Doc. So why is there so much ado about nothing? Frenkel, and John F.
The Reference Frame: John Baez, octonions, and string theory
May 3, at 7: April 29, at At any moment in time a string is a one-dimensional thing, like a curve or line. So, if string theory is right, the octonions are not a useless curiosity: Hamilton ‘s discovery of quaternions. Posted here because answering is disabled in: It’s not really demonstrably wrong – unlike Garrett Lisi’s pseudoscientific “theories of everything” that can’t agree with the most elementary facts of particle physics such as parity violation.
The automorphism group of the integral octonions is the group G 2 F 2 of orderwhich has a simple subgroup of index 2 isomorphic to the unitary group 2 A 2 3 2. Because then its vibrations can be described using numbers in a division algebra.
In other projects Wikimedia Commons. April 30, at 3: So when John Huerta finally made it really clear, it seemed worth explaining — in detail in some math papers, and in a popularized way octoniions Scientific American. Stevenson, Projective PlanesW. There might be a reason that octonions have not made it into physics in years. Supersymmetric world from a conservative viewpoint.
And as usual those who are stuck in the investment till their necks will not be the first ones to blow the whistle that a market correction is coming. Hans Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, mimeographed notes, April 29, at 9: I never understood the big deal with octonions.
Springer and Ferdinand D. By way, I just won a case of scotch from Dave Ring: SSSR Wiskunde 8 Bulletin of the American Mathematical Society.
Proudly powered by WordPress. Apparently the nonassociativity messes things up!
In particular, John Huerta has done a bunch of calculations showing that the trick relating octonions to strings in 10 dimensions and 2-branes in 11 dimensions does not octoniond to work one dimension higher. Views Read Edit View history. Ingram, Cambridge University Press, Cambridge,pp. A Course for Mathematicians2 volumes, Amer.
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They are noncommutative and nonassociativebut satisfy a weaker form of associativity; namely, they are alternative. Corinne Manogue, Tevian Dray, and others wrote some very illuminating papers where they spell out the properties of octonions and postulate about their applications to physics: Coincidentally string theorists have for years been saying that only dimensional versions of the theory are self-consistent.
Each pair of distinct points lies on a unique line and each line runs through basz three points. The existence of octoniohs norm on O implies the existence of inverses for every nonzero element of O.
Mathematics > Rings and Algebras
We need better Physics. The octonions play a significant role in the classification and construction of other mathematical entities.
Michael Nielsen, an achieved quantum information researcher, just wrote a fun essay What does the quantum state mean? Check Mathematica on Abez.
Which brings us back to your quote. Even Bott periodicity uses the octonions in disguise to cycle through the other three division algebras. Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H. All the cancellations that Baez et al.