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expanded L-infinity-algebra as indicated on the nCafe, here
I have finally added the statement that every $L_\infty$-algebra is quasi-isomorphic to a dg-Lie algebra (in char 0), together with a pointer to Kriz-May. Also at dg-Lie algebra
Urs, do you have any thoughts in the parallel discussion on the very related question I opened under Magnus infinity algebra ?
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 $L_\infty$-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 $n$Lab entry, for the sake of those students coming after him…
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.
Okay, I have filled in the signs of the “$L_\infty$-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 $L_\infty$-algebra structure on $\mathfrak{g}$ is the same as a dg-algebra structure on $\wedge \mathfrak{g}^*$ (for the degreewise finite dimensional case).
I have substantially polished the section Definition – In terms of higher brackets.
Jim Stasheff suggested that the entry L-infinity algebra ought to contain discussion of literature of $L_\infty$-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.
Together with Jim Stasheff (via email) we have been further expanding the list of references for $L_\infty$-algebras in physics.
For ease of editing and linking, I have now split it off as a separate entry L-infinity algebras in physics.
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.
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 $L_\infty$-algebras.
added a remark (here) amplifying the ind-conilpotency of $CE_\bullet(\mathfrak{g})$
added references Buijs-Felix-Murillo 12 and Buijs-Murillo 12 on $L_\infty$-algebras as models for rational homotopy theory
The statement of the strong homotopy Jacobi identity contains some errors. Note that there are $n$ $v$’s in the equation. This is correct, since a composition of $l_j$ with $l_i$ should take $n=i+j-1$ arguments. For this reason, it should say “for all $n$-tuples” (not $n+1$). I believe the inner sum should be over $(i,j-1)$-unshuffles, since $(i,j)$-unshuffles would permute $n+1$ 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.
I don’t have the leisure to check now. But if/where it has “$n+1$” when it should read “$n$” and you are sure you know what you are doing: Please just fix it! (Just hit “edit” at the bottom of the page).
Looks good. Thanks!
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