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Added these pointers:
Daniel K. Biss, Daniel Dugger, Daniel Isaksen, Large annihilators in Cayley-Dickson algebras, Communications in Algebra 36 (2), 632-664, 2008 (arxiv:math/0511691)
Daniel K. Biss, Daniel Christensen, Daniel Dugger, Daniel Isaksen, Large annihilators in Cayley-Dickson algebras II, Boletin de la Sociedad Matematica Mexicana (3) 13(2) (2007), 269-292 (arxiv:math/0702075)
Daniel K. Biss, Daniel Christensen, Daniel Dugger, Daniel Isaksen, Eigentheory of Cayley-Dickson algebras, Forum Mathematicum 21(5) (2009), 833-851 (arxiv:0905.2987)
added pointer to the original article:
For the record and for my peace of mind, I have spelled out in full detail the computation which shows the equivalence between the component formula
$(a,b) (c, d) \;\coloneqq\; (a c - d \overline{b}, \overline{a} d + c b)$and the relations
$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} (a \ell) b = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell) = - \overline{a b}$As it goes, this showed that the previous version of the entry was wrong: There are two versions of the component formula, depending on whether one identifies
$(a,b) \leftrightarrow a + \ell b$or
$(a,b) \leftrightarrow a + b \ell$and the previous version wasn’t consistent about this across subsections.
I have now changed everything to the first version. For completeness one should eventually add at least a remark about the second version.
Who is the first to state the minimal set of relation
$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} (a \ell) b = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell) = - \overline{a b}$?
Around (6) of Dickson 1919 the idea of generators $i, j, k , \ell$ appears, but not this minimal choice of set of relations.
Is this original to Baez 02, where it appears in the second half of section 2.2?
Where “split variants” are mentioned, I added explicit pointer to
Added a few words about how the Cayley-Dickson algebras are better understood as higher algebras instead of as 1-algebras with fewer and fewer “nice” properties.
From this point of view there is an interesting path to be explored: given that exceptional Lie group such as G2 arise as automorphism groups of the octonions, one can perhaps discover higher exceptional groups (a term that I was not able to find in the literature) as the higher automorphism groups of these higher algebras and of whatever the stacky generalization of projective space is for these. (And of course, given the relation between the first few Cayley-Dickson real algebras and spin group representations, perhaps there is an analogous relation between the further algebras regarded as higher algebras, and the other groups of the O(n) tower, e.g. the string 2-group. Hopefully this can be sorted out some day.)
This is a good point to add.
But what’s the general statement for higher cocycles? Your addition starts out saying that:
Albuquerque & Majid describe the Cayley-Dickson process […] as $\mathbb{Z}_2^n$ group algebras over the reals twisted by a group $n$-cocycle.
but looking at their article, I spot only 2- and 3-cocycles.
In particular their reformulation of the Cayley-Dickson construction in their Prop. 4.1 is all in terms of 2-cochains, no?
That’s right, that paper is only concerned with 2- and 3-cocycles, but I thought the generalization to n-cocycles was too obvious not to attribute it to this paper. Though now that you mention it, looking for the words “sedenions” and “4-cocycle” online does not seem to return any results…
I can see how you sensed the beginning of a suggestive pattern.
But there are several suggestive-looking patterns which all famously break down at or after the octonions that some caution may be in order.
If there is indeed no known continuation to higher cocycles as your text indicates, then better to rewrite it accordingly.
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