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In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's):

Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative algebra $A$ has a natural structure of an algebra over a chain operad of the little 2-disks operad $E_2$.

("Chain operad of $E_2$" means any operad in $Ch$ whose homology is isomorphic to the homology of $E_2$, i.e. to the Gerstenhaber operad.)

Various versions of Deligne's conjecture have been proven, e.g. by Gerstenhaber-Voronov, Tamarkin, Voronov, McClure-Smith (see also 1, 2), and Kontsevich-Soibelman.

Now, in some situations, $HH^*(A)$ can be equipped with a BV operator $\Delta$ compatible with the Gerstenhaber algebra structure, so that it becomes a BV algebra. For instance, this was proven by Tradler in the case that $A$ is unital and equipped with a symmetric, invariant, and non-degenerate inner product. And BV algebras are the same thing as algebras over the framed little 2-disks operad $\tilde E_2$. So...

Question: With suitable hypotheses on $A$, can $CC^*(A)$ be given the structure of an algebra over a chain operad of $\tilde E_2$?

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Yes, this is the "cyclic Deligne conjecture", which also has several proofs by now. I believe the first one was Kaufmann's, using the cacti operad.

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  • $\begingroup$ Thanks! Do you know whether it is expected (proven or not) that $CC^*(A)$ has any more structure? $\endgroup$ – Nathaniel Bottman Jun 23 '16 at 16:51
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    $\begingroup$ I would believe not. $\endgroup$ – Dan Petersen Jun 23 '16 at 16:53
  • $\begingroup$ Sorry for cutting in long after the conversation, but it seems that Kaufmann assumes A has inner product, and then works with usual Hochschild cochains. What if A is just associative and one works with cyclic cochains? $\endgroup$ – Vladimir Baranovsky Oct 11 '16 at 21:49

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