Timeline for Explicit $L_\infty$-operations on Hochschild cochains of $A_\infty$-algebra
Current License: CC BY-SA 4.0
6 events
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| Sep 20, 2021 at 12:34 | comment | added | Jack Smith | 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. | |
| Sep 20, 2021 at 6:06 | comment | added | Stefan Waldmann | 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. | |
| Sep 19, 2021 at 19:25 | comment | added | Jack Smith | 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 $\mathrm{CC}^*$? | |
| Sep 17, 2021 at 14:01 | comment | added | Stefan Waldmann | 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. | |
| S Sep 17, 2021 at 13:57 | review | First questions | |||
| Sep 17, 2021 at 14:36 | |||||
| S Sep 17, 2021 at 13:57 | history | asked | Jack Smith | CC BY-SA 4.0 |