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nLab > Latest Changes: Lie algebra weight system

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2019

    starting something – not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2019

    now some minimum in place

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2019

    added graphics illustrating how a Lie algebra weight system reads in a horizontal chord diagram and spits out a string diagram

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2019

    made a feeble attempt to make the graphics display also the kind of string diagram assigned to a chord/Jacobi diagram (here)

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 30th 2019

    Given that you mentioned Lie algebra objects among the references, Jacob Lurie gave a great seminar recently on Lie algebras and homotopy theory, and advertised upcoming work of Heuts on a ’torsion-sensitive’ version of the result about pointed simply-connected rational spaces being equivalent to rational dg-Lie algebras.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2019

    David, this should really be in another thread. You should add that reference to the relevant nLab entry (not to “Lie algebra weight system”, though, unless you make some actual relation to that). If you add a comment below the edit box there, the thread you want will get started here automagically.

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 30th 2019

    OK. (late here, not going to sort this out now)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2019

    David, maybe to clarify in case it’s not clear: There is nothing deep to be sorted out, it’s straightforward:

    We have dedicated threads for topics here, so that people can find and participate in the relevant discussion on the topics of specific pages. Even though, for the time being, I am alone in speaking about Lie algebra weight systems here, this need to remain so in all eternity. I hope.

    You may not care about Lie algebra weight systems, but in order to allow potential other contributors to interact here we must keep the discussion on topic. I hope I am not being pedantic here, this seems to be basic netiquette.

    This should also be in your own interest: Neither the topic of Lie algebra weight systems nor the reference that you want to talk about in #5 are being being done a favour by randomly conflating their discussion here. Because – as far as I can tell – they are not related at all beyond some super general umbrella of Lie theory, right?

    But the good thing is that there is also no need to save space this way! With the exact number of keystrokes that you used to produce #5 and #7 you can put your reference into ts proper place and have a dedicated discussion thread on it started automatically by our software, with the invaluable bonus that now people who may be interested in your topic get to see it in the Forum software and may find it through nLab searches.

    So: just drop your reference in its proper place, maybe in the references section at rational homotopy theory, add a word of remark in the comment box (that’s in total less than two dozen keystrokes) and voila: all sorted out and everything in place!

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 2nd 2019

    Urs,

    please don’t take my post as evidence of ignorance, but of poorly expressing the intent of the comment. The mention of the work of Heuts was meant to be an aside, and the point was that Lurie gave a nice intro to Lie algebra objects, and I thought it might be interesting to think about your recent work looking at weight systems/braids in light of this. Also in light of the question here which someone else asked in relation to detail about the universal category with a Lie algebra object, also mentioned in Lurie’s talk. If the weight systems/braids/etc have something sensible to say about this universal case it could be interesting.

    When Heuts’ work actually appears (as a talk or preprint) I will definitely add it in the appropriate place.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019
    • (edited Dec 2nd 2019)

    Not sure how it is we are talking past each other, but I have moved your comment to the thread on Lie algebra objects: (here).

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 2nd 2019

    OK, thanks. Easier in person :-)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019
    • (edited Dec 2nd 2019)

    I have added (here) statement of the main theorem of Bar-Natan 96, saying that for weight systems on horizontal chord diagrams it is true that they all come from Lie algebras, in fact any one special linear Lie algebra will do.

    I will also give this its dedicated Theorem-entry now: all horizontal weight systems are Lie algebra weight systems

    diff, v10, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019

    starting a section As expectation values of single trace observables (here) on how Lie algebra weight system values are the coefficient in the Wick-expansion of expectation values of single trace observables.

    Just a start, not cleaned up yet, but need a break.

    diff, v11, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019

    added graphics (here) showing how the “traced Wick theorem” makes single trace observables be Lie algebra weight systems.

    diff, v14, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    added references for this observation:

    The observation that round chord diagrammatics controls the contractions in expectation values of single trace observables subject to Wick’s theorem appears (specifically in disucssion of SYK model-like systems and without mentioning of weight systems) in these articles (see also at weight systems on chord diagrams in physics):

    • Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Section 2.2 Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N 21/N^2, JHEP 04 (2018) 146 (arXiv:1801.02696)

    • Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large NN expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (arXiv:1806.03271)

    • Micha Berkooz, Prithvi Narayan, Joan Simón, Section 2.1 of Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)

    • Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 2 of: Towards a full solution of the large NN double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)

    • Vladimir Narovlansky, Slides 5 to 21 of: Towards a Solution of Large NN Double-Scaled SYK, 2019 (NarovlanskySYK19.pdf:file)

    diff, v15, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2020

    added pointer to

    (thanks to David C. for finding this!)

    diff, v27, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2021

    added this pointer:

    diff, v34, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2021

    added pointer to:

    diff, v34, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2021

    added pointer to:

    • Fumikazu Nagasato, An Exposition of the 𝔰𝔩(N,ℕ)\mathfrak{sl}(N,\mathbb{N})-Weight System, Journal of Lie Theory 17 (2007), No. 2, 263–281, Heldermann Verlag 2007 (jlt:17014)

    diff, v36, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 22nd 2024

    added pointer to today’s

    • Dmitry Khudoteplov, Elena Lanina, Alexey Sleptsov: Construction of Lie algebra weight system kernel via Vogel algebra [arXiv:2411.14417]

    diff, v39, current

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