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

TL;DR

This paper introduces Hoch-algebras, a new category of associative algebras with an extra operation satisfying Hochschild cocycle relations, and establishes their structural properties and categorical equivalences.

Contribution

It defines Hoch-algebras with a magmatic operation satisfying Hochschild cocycle relations and proves the triples of operads are good, establishing an equivalence with Hoch-bialgebras.

Findings

The free Hoch-algebra is described via planar rooted trees.

The triples (As, Hoch, Mag^ Infty) are shown to be good.

An equivalence between connected infinitesimal Hoch-bialgebras and Mag^ Infty-algebras is established.

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.