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nLab > Latest Changes: division algebra article

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    • CommentRowNumber1.
    • CommentAuthorrschwieb
    • CommentTimeOct 27th 2020
    I think some editing to the division algebra article is necessary (https://ncatlab.org/nlab/show/division+algebra)

    The approach taken has been presented with a few inconsistencies. It appears to follow John Baez's article (partially) in defining it as a (possibly nonassociative) algebra over a field with no nontrivial zero divisors. This would be fine if the entire article assumed finite dimensionality, but the first paragraph does not, allowing something like R[x] to be termed "a division algebra."

    While I understand the approach in the Baez article is coherent and just fine, I have to question whether or not "no nontrivial zero divisors" is right the way to present division algebras in this context.

    In wikipedia, for example, it's defined just by saying "every element has a two-sided inverse," which Baez calls an algebra "with multiplicative inverses." Baez's definition is apparently strictly weaker.

    I don't know which approach is more historically accurate and/or considered 'the right' approach in current theory.

    I would think that, if not "algebra with multiplicative inverses", then a definition saying that $ax=b$ and $xa=b$ both have unique solutions for any b and any nonzero a, would be the right way to go.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2020
    • (edited Oct 27th 2020)

    Hi,

    thanks for joining in.

    The article division algebra is not more than a stub. The few lines it does feature hardly constitute an “approach” to anything, but are just some words to fill the void between the entry title and the equally minimalistic “list” of references.

    Which means:

    There is no crowd of editors here needing to be convinced to change their ways – instead there is a dearth of people taking care of this entry.

    You could be the first. If you do have the energy to write about division algebras, then please go ahead and make this entry yours!

    Just hit “edit” at the bottom of the entry. Syntax is fairly straightforward; and no need to worry about formatting too much for the beginning, once you are at it, regulars here are likely to lend a hand with the formatting.

    Hints on writing here on the nForum:

    To hyper-link to an entry, just include it in square brackets, as in

      [[division algebra]]

    To make maths-formulas coded in between dollar signs appear as intended, choose “Markdown+Itex” below the edit pane here.

    Beware that our Instiki has the “feature” that it renders consecutive letters in maths-mode in \mathrm, so for products to come out as intended, give them a whitespace as in $a x = b$ instead of $ax=b$.

    • CommentRowNumber3.
    • CommentAuthorrschwieb
    • CommentTimeOct 27th 2020
    Thanks for all the advice and encouragement. This is indeed different from my previous experince at wikipedia.

    But I really needed to ask around, because I don't really know what an equitable solution will be...

    I lean toward the "unique solution" description, but with no real fluency in the literature.
    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 27th 2020

    I would maybe prefer the definition at https://encyclopediaofmath.org/wiki/Division_algebra over Wikipedia.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2020

    Just hit “edit” at the bottom of the page and go ahead.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 28th 2020

    I leave it to rschwieb, who has in interest in this, it seems.

    My dissatisfaction with the article is that it states certain things are easy to construct but gives no reference or construction, and I have no idea how to construct them!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2020

    That statement was added by John Baez in rev 1, so it’s probably in his review.

    I’ll suggest that anyone who is actually interested in division algebras in general and in their nnLab entry in particular turn from making a fuss here on the forum and just do the little bit of literature search and the few keystrokes that it takes to add a decent paragraph or two to the entry. This is not rocket science, is it.

    • CommentRowNumber8.
    • CommentAuthorrschwieb
    • CommentTimeNov 9th 2020
    I took a crack at it. Thanks fellas.
    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 10th 2020

    May I strongly suggest that “there exist” statements about algebras with certain properties are either given citations to papers or books with the description, or, better, these descriptions are included in the page (with a citation). At present I have no feeling for what these algebras could be or how to construct them.

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