TITLE: Analogy of Gerstenhaber algebra
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.