TITLE: bialgebra cohomology
QUESTION [5 upvotes]: It seems that the Gerstenhaber-Schack (bi)complex of an associative bialgebra carries a homotopy e_3-algebra structure and a degree 2 Lie algebra bracket, up to homotopy. Does anyone know about a reference where these structures have been explicitly shown?

REPLY [2 votes]: It's done in Ginot-Yalin Deformation theory of bialgebras, higher Hochschild cohomology and formality (arXiv:1606.01504

More precisely, my undertsanding is that what's true by abstract non-sense is that under some (co)nilpotency condition there's an equivalence between $E_2$-algebras and (dg-)bialgebras, and that the deformation complex of an $E_2$-algebra is $E_3$. What is done in that paper apart from carefully writing this up, is identifying this with the standard GS complex explaining and precisely in what way t controls deformations of bialgebra in this setting.