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• CommentRowNumber1.
• 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

• CommentRowNumber2.
• CommentTimeAug 2nd 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…

• CommentRowNumber3.
• CommentTimeAug 2nd 2021

• 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

• 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 $H^4(B G)\to H^4(B K)$, of the inclusion map $\pi_3(K)\to \pi_3(G)$ (both associated to a subgroup inclusion $K\hookrightarrow G$) and so on.

• CommentRowNumber6.
• 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 $f$ of an algebra $\tilde{G}$ into $G$ via $\tilde{X}\rightarrow f(\tilde{X}) \in G$?

Would the restriction map of the inclusion map as shown above just be a deformation retraction from $G$ to $K$?