Hochschild two-cocycles and the good triple $(As,Hoch,Mag^\infty)$

Abstract

Hochschild two-cocycles play an important role in the deformation \`a la Gerstenhaber of associative algebras. The aim of this paper is to introduce the category of Hoch-algebras whose objects are associative algebras equipped with an extra magmatic operation \succ verifying the Hochschild two-cocycle relation: (x \succ y)*z+ (x*y)\succ z= x\succ (y*z)+ x*(y\succ z). The free Hoch-algebra over a K-vector space is given in terms of planar rooted trees and the triples of operads (As,Hoch, Mag^\infty) endowed with the infinitesimal relations are shown to be good. We then obtain an equivalence of categories between connected infinitesimal Hoch-bialgebras and Mag^\infty-algebras.