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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.
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.
OK. (late here, not going to sort this out now)
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!
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.
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).
OK, thanks. Easier in person :-)
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
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 $1/N^2$, JHEP 04 (2018) 146 (arXiv:1801.02696)
Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large $N$ 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 $N$ double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)
Vladimir Narovlansky, Slides 5 to 21 of: Towards a Solution of Large $N$ Double-Scaled SYK, 2019 (NarovlanskySYK19.pdf:file)
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