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    • 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)(acdb¯,a¯d+cb) (a,b) (c, d) \;\coloneqq\; (a c - d \overline{b}, \overline{a} d + c b)

    and the relations

    a(b)=(a¯b),AA(a)b=(ab¯),AA(a)(b)=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+b (a,b) \leftrightarrow a + \ell b


    (a,b)a+b (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(b)=(a¯b),AA(a)b=(ab¯),AA(a)(b)=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,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?

    diff, v27, current