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nLab > Latest Changes: Dynkin diagram

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  1.  
    • CommentRowNumber1.
    • CommentAuthorGrant_Bradley
    • CommentTimeJul 6th 2021

    Added definition of dynkin diagram and the dynkin index for now. will add significance of this index in 4-D gauge theories and instanton physics

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorGrant_Bradley
    • CommentTimeAug 3rd 2021

    Added brief relation of the dynkin index to embeddings of a gauge field, specifically the topological charge. Inserted placeholder for proof until I can find the original journal article…

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorGrant_Bradley
    • CommentTimeAug 3rd 2021

    add pointer to reference

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 3rd 2021
    • (edited Aug 3rd 2021)

    I have re-arranged slightly:

    • Re-ordered the references (in an entry on Dynkin diagrams a reference on application of Dynkin indices to mathematical physics should probably not precede references on the maths of Dynkin diagrams).

      In fact, probably Dynkin index should be a separate entry altogether.

    • removed the empty “Proof” subsection: Proofs should not be subsections but proof environments (see at HowTo here). Also, there is no discernible theorem stated yet, so it seems out of place to announce a proof. Once you have a theorem statement and really go about writing the proof, just type \begin{proof}... \end{proof}

    • added a direct link from the text to your reference Myers, de Roo & Sorba 1979. That looks like a good reference, thanks for adding it. Maybe all you really mean to do here is to point the reader to there for more, which is worthwhile. For that a Remark environment seems appropriate, and so I have made one (here).

    • changed the broken pointer to “topological charge” to one to instanton number

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 3rd 2021
    • (edited Aug 3rd 2021)

    A lot of the Dynkin index material is lifted from here. I agree it should go in its own entry, and linked to concepts like the index of the image of the restriction map H 4(BG)→H 4(BK)H^4(B G)\to H^4(B K), of the inclusion map π 3(K)→π 3(G)\pi_3(K)\to \pi_3(G) (both associated to a subgroup inclusion K↪GK\hookrightarrow G) and so on.

    • CommentRowNumber6.
    • CommentAuthorGrant_Bradley
    • CommentTimeAug 3rd 2021
    • (edited Aug 3rd 2021)

    Thank you for the pointers. Yes, I used the Dynkin index entry in the SUSY encyclopedia (listed in references) as a starting point.

    Is the index of the image of the restriction map above related to a faithful embedding, defined by some injective map ff of an algebra G˜\tilde{G} into GG via X˜→f(X˜)∈G\tilde{X}\rightarrow f(\tilde{X}) \in G?

    Would the restriction map of the inclusion map as shown above just be a deformation retraction from GG to KK?

1 to 6 of 6

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