We know that the (endomorphism) operad structure on the cochain complex of an associative algebra induces a Gerstenhaber algebra structure on the cohomology. My query is: if a cochain complex has a (endomorphism) dioperad structure, what structure should we expect in the cohomology? Is there an analogy of Gerstenhaber algebra in this case?
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One possible answer is contained in the paper of Victor Ginzburg and Travis Schedler, "Free products, cyclic homology, and Gauss-Manin connection", https://arxiv.org/abs/0803.3655. You will be in particular interested in Section 7.2 there, where the structure that naturally arises on the chain level is discussed. This structure also appears in my recent work with Sergey Shadrin and Bruno Vallette https://arxiv.org/abs/1510.03261, see in particular Section 3.1 of that paper.
answered May 3, 2017 at 21:25
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$\begingroup$ Thank you @Vladimir, for the reference. I have one more point here: The dioperad structure in the cochain level, in my case, arose from a bialgebra structure, whereas, the structure in the paper you cited does not make use of the co-product at all. $\endgroup$– aniCommented May 5, 2017 at 8:56
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$\begingroup$ @ani maybe you can somehow clarify the question - you mean you are still using the standard Hochschild complex, but for a bialgebra, and then use the bialgebra structure to define a dioperad? it would be beneficial to have some clear statements in place of "if a cochain complex has a (endomorphism) dioperad structure"... $\endgroup$ Commented May 5, 2017 at 14:34