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nLab > Latest Changes: Jordan superalgebra

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 3rd 2020

    Since I found out there is such a thing, I’d better start a page.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 26th 2020

    (I’m still going through a backlog of old nForum posts.) I presume Jordan algebras are not algebras of a traditional permutative operad? Because if they were, then I’d wonder whether the Jordan superalgebras could be defined as algebras of such an operad, but valued in superspaces.

    • CommentRowNumber3.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 16th 2023
    • (edited Feb 16th 2023)
    I thought once I figured out how to express the Jordan algebra identities in a way that showed there was an operad for Jordan algebras. The tough part is the identity

    (xy)(xx) = x(y(xx))

    I think the trick is to "polarize" it, sort of like how you can use polarization to get from

    xx = 0

    to

    xy = - yx

    (which is just as strong except in characteristic 2). You can polarize cubic forms and get trilinear forms.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2023

    have adjusted the wording in the Idea-section (here):

    by linking to supercommutative superalgebra the entry can omit the distraction of trying to recall this standard notion, freeing room for the important aspects, namely the “Lie like-product” and the “super Jordan identity” – maybe somebody wants to state these.

    diff, v11, current

1 to 4 of 4

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