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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 19th 2009

    expanded L-infinity-algebra as indicated on the nCafe, here

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2011

    I have finally added the statement that every L L_\infty-algebra is quasi-isomorphic to a dg-Lie algebra (in char 0), together with a pointer to Kriz-May. Also at dg-Lie algebra

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeAug 8th 2011
    • (edited Aug 8th 2011)

    Urs, do you have any thoughts in the parallel discussion on the very related question I opened under Magnus infinity algebra ?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2011
    • (edited Sep 11th 2011)

    I have changed at L-infinity algebra the “skew symmetric” brackets to “graded-symmetric”, for clarity.

    I am having an email exchange with somebody trying to learn the basics of L L_\infty-algebras. This highlights what is of course evident: the whole discussion here could do with more examples and more details.

    I don’t have the time to look into this. But maybe I can convince my correspondent to include discussion of his newly gained insights into the nnLab entry, for the sake of those students coming after him…

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeSep 12th 2011
    @urs: I have changed at L-infinity algebra the "skew symmetric" brackets to "graded-symmetric", for clarity.

    I don't have time to check whether you are writing at the level of the underlying gr vector space
    or at the Chevalley Eilenberg shifted level
    graded symmetric is fine after the shift
    graded skew-symmetric at the unshifted level
    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2011
    • (edited Sep 12th 2011)

    Since it says “ +\mathbb{N}_+-graded” it’s shifted, hence “graded-symmetric” is correct.

    Maybe I find the time to expand and polish the whole entry. Or maybe somebody else does.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2011

    Okay, I have filled in the signs of the “L L_\infty-Jacobi identity” and then added a further section

    Reformulation in terms of semifree dg-algebra – Details with an explicit pedestrian walk through the proof that an L L_\infty-algebra structure on 𝔤\mathfrak{g} is the same as a dg-algebra structure on 𝔤 *\wedge \mathfrak{g}^* (for the degreewise finite dimensional case).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2015

    I have substantially polished the section Definition – In terms of higher brackets.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2016
    • (edited Jan 12th 2016)

    Jim Stasheff suggested that the entry L-infinity algebra ought to contain discussion of literature of L L_\infty-algebras appearing in physics. I gather he would like to use such a list for his talk at “Higher structures in geometry and physics” in Bonn next week. Since we didn’t have such a list yet, I have now added one. Naturally the examples that come to my mind tend to be those that I have worked on myself. Please feel invited to add more.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2016

    Together with Jim Stasheff (via email) we have been further expanding the list of references for L L_\infty-algebras in physics.

    For ease of editing and linking, I have now split it off as a separate entry L-infinity algebras in physics.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016

    Here are now the slides from Jim’s talk yesterday: Stasheff 16.

    Jim asks me to add that a more complete preprint is in preparation, and that meanwhile comments on the slides are most welcome.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2017

    I have polished a little the section Definition – In terms of higher brackets, at L-infinity algebra.

    And in the section In terms of semifree differential coalgebra I have added text at the beginning highlighting that this reformulation was shown already in the original articles on L L_\infty-algebras.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2017

    added a remark (here) amplifying the ind-conilpotency of CE (𝔤)CE_\bullet(\mathfrak{g})

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018

    added references Buijs-Felix-Murillo 12 and Buijs-Murillo 12 on L L_\infty-algebras as models for rational homotopy theory

    diff, v90, current

  1. I have added the reference for the survey posted today on arxiv by Stasheff, but am not sure if the formatting (that I used in editing this article) is appropriate.


    diff, v94, current

    • CommentRowNumber16.
    • CommentAuthorGuest
    • CommentTimeSep 27th 2019
    I'm unsure as I'm using this page to learn what a L-infinity algebra is, but shouldn't the definition 3.2 "In terms of higher brackets" of a L-infinity algebra use a n-tuple and a (i,n-i) unshuffle instead of a (n+1)-tuple and a (i,j) unshuffle?
    • CommentRowNumber17.
    • CommentAuthormrmuon
    • CommentTimeNov 26th 2019

    The statement of the strong homotopy Jacobi identity contains some errors. Note that there are nn vv’s in the equation. This is correct, since a composition of l jl_j with l il_i should take n=i+j1n=i+j-1 arguments. For this reason, it should say “for all nn-tuples” (not n+1n+1). I believe the inner sum should be over (i,j1)(i,j-1)-unshuffles, since (i,j)(i,j)-unshuffles would permute n+1n+1 objects.

    A similar error occurs when Loday and Valette try to state this identity for the operad. I am actually struggling to find the correct identity written down with signs anywhere.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2019

    I don’t have the leisure to check now. But if/where it has “n+1n+1” when it should read “nn” and you are sure you know what you are doing: Please just fix it! (Just hit “edit” at the bottom of the page).

    • CommentRowNumber19.
    • CommentAuthormrmuon
    • CommentTimeNov 26th 2019

    I corrected the aforementioned error. This is my first edit, so I hope I haven’t broken anything.

    diff, v97, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2019

    Looks good. Thanks!

    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 24th 2021
    • (edited Jul 24th 2021)

    etc (where the \vee is just a funny way to write the wedge \wedge, in order to remind us that:…)

    remind us what, exactly?

    work out or see the references for the signs and prefacors

    better to see this on the actual page

    exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below