HOCHSCHILD TWO-COCYCLES AND THE GOOD TRIPLE (As, Hoch, M ag∞)

Abstract

The aim of this paper is to introduce the category of Hoch-algebras whose objects are associative algebras equipped with an extra magmatic operation ≻ verifying the following relation motivated by the Hochschild two-cocycle identity: R2 : (x ≻ y) ∗ z + (x ∗ y) ≻ z = x ≻ (y ∗ z) + x ∗ (y ≻ z). Such algebras appear in mathematical physics with ≻ associative under the name of compatible products. Here, we relax the associativity condition. The free Hoch-algebra over a K-vector space is then given in terms of planar rooted trees and the triple of operads (As, Hoch, Mag∞) endowed with the infinitesimal relations is shown to be good. Hence, according to Loday’s theory, we then obtain an equivalence of categories between connected infinitesimal Hochbialgebras and Mag∞-algebras.