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

Year 2011, Volume: 10 Issue: 10, 76 - 84, 01.12.2011

Philippe Leroux

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.

Keywords

Hoch-algebras, infinitesimal Hoch-algebras, magmatic algebras, good triples of operads, cocycles d’Hochschild

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