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It is well-known that the Hochschild cochain complex $\mathrm{CC}^*(A)$ of an associative algebra $A$ carries a lot of structure. In particular: a differential, a cup product, and a bracket, which make the Hochschild cohomology $\mathrm{HH}^*(A)$ into a Gerstenhaber algebra. For concretenes, I have in mind that we're using the standard bar complex model for $\mathrm{CC}^*$ throughout.

If $A$ is now an $A_\infty$-algebra instead, the differential and cup product on $\mathrm{CC}^*(A)$ become the first two operations of an $A_\infty$-structure. There are neat explicit formulae for the higher operations written down in lots of places. There is also a nice formula for the bracket, and I understand that it fits into an $L_\infty$-structure (or, more fancily, a $G_\infty$-structure) on $\mathrm{CC}^*(A)$. However, I have not been able to find explicit formulae for the higher $L_\infty$-operations written down anywhere. I have only been able to find more abstract discussions, eg at the level of operads. I believe that in principle one could extract formulae from these, but this is a bit beyond me at present! Similarly, one could presumably get formulae by strictifying $A$ to a dg-algebra, but this seems like it would get messy.

Question 1. Are there explicit formulae for the higher $L_\infty$-operations on $\mathrm{CC}^*(A)$, when $A$ is an $A_\infty$-algebra?

Question 0. Are these higher operations in fact all zero? There doesn't seem to be an obvious way to generalise the formula for the $\mathfrak{l}_2$ operation to higher $\mathfrak{l}_k$, and if the higher operations all vanished then it would explain why I've failed to find them written down :)

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    $\begingroup$ I'm not sure if this is what you want but the Gerstenhaber bracket is a (graded) Lie bracket on the Hochschild complex (not yet a compelx) for every module $V$ directly. It needs no structure on $V$ at all and satisfies the Jacobi identity strictly without higher homotopies. $\endgroup$
    – Stefan Waldmann
    Commented Sep 17, 2021 at 14:01
  • $\begingroup$ Thanks @StefanWaldmann - is $V$ an $A_\infty$-module over $A$ in your comment? What I really want to know is whether the differential and bracket (with vanishing higher operations) give the correct $L_\infty$-structure on $\mathrm{CC}^*(A)$. E.g. if you replaced $A$ with a quasi-equivalent dg-algebra, would this result in a quasi-equivalent $L_\infty$-structure on $\mathrm{CC}^*$? $\endgroup$
    – Jack Smith
    Commented Sep 19, 2021 at 19:25
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    $\begingroup$ Oh, sorry: I should have used your notation: $V = A$. The point is that the Gerstenhaber bracket only depends on the underlying linear structure of your $A$, not on any sort of algebraic structure. So the question is perhaps how to encode additional structures like $A_\infty$ on your $A$ in the Hochschild complex. This results in e.g. the Hochschild differential plus additional stuff, but the Gerstenhaber bracket was already there from the beginning of time. $\endgroup$
    – Stefan Waldmann
    Commented Sep 20, 2021 at 6:06
  • $\begingroup$ Ah, great, I see now, thanks! It seems like there really should not be any higher $L_\infty$-operations then. I'm not sure how to check this (i.e. check that the resulting $L_\infty$-algebra quasi-isomorphism type of $\mathrm{CC}^*(A)$ is an $A_\infty$-algebra quasi-isomorphism invariant of $A$), since functoriality of Hochschild cochains under $A_\infty$-algebra quasi-isomorphisms is a bit messy, so I will leave the question open in case anybody else has suggestions for this. $\endgroup$
    – Jack Smith
    Commented Sep 20, 2021 at 12:34

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