Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched etcs fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 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…

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorGrant_Bradley
    • CommentTimeAug 2nd 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 KGK\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?

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)