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nLab > Latest Changes: Cayley-Dickson construction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2018

    made explicit the alternative form of the definition (here) in terms of adjoining a generator and imposing relations

    diff, v17, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2019

    Added these pointers:

    diff, v20, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2020

    fixed the second relation in the definition of the CD-double via generators-and relations

    diff, v23, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2020

    added pointer to the original article:

    • Leonard Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem, Annals of Mathematics Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)

    diff, v25, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2020

    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)≔(ac−db¯,a¯d+cb) (a,b) (c, d) \;\coloneqq\; (a c - d \overline{b}, \overline{a} d + c b)

    and the relations

    a(ℓb)=ℓ(a¯b),AA(aℓ)b=(ab¯)ℓ,AA(ℓa)(bℓ)=−ab¯ 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)↔a+ℓb (a,b) \leftrightarrow a + \ell b

    or

    (a,b)↔a+bℓ (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.

    diff, v25, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2020

    Who is the first to state the minimal set of relation

    a(ℓb)=ℓ(a¯b),AA(aℓ)b=(ab¯)ℓ,AA(ℓa)(bℓ)=−ab¯ 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,ℓi, 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?

    diff, v27, current

    • CommentRowNumber7.
    • CommentAuthorperezl.alonso
    • CommentTimeNov 2nd 2023

    Albert’s generalization of Cayley-Dickson

    diff, v31, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2023

    Where “split variants” are mentioned, I added explicit pointer to

    diff, v32, current

    • CommentRowNumber9.
    • CommentAuthorperezl.alonso
    • CommentTimeNov 23rd 2023

    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.)

    diff, v33, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2023
    • (edited Nov 23rd 2023)

    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 ℤ 2 n\mathbb{Z}_2^n group algebras over the reals twisted by a group nn-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?

    diff, v34, current

    • CommentRowNumber11.
    • CommentAuthorperezl.alonso
    • CommentTimeNov 23rd 2023

    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…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2023

    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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