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

Authors : Philippe Leroux

Pages : 76-84

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Publication Date : 2011-12-01

Article Type : Research Paper

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 : insert ignore into journalissuearticles values(x ≻ y); ∗ z + insert ignore into journalissuearticles values(x ∗ y); ≻ z = x ≻ insert ignore into journalissuearticles values(y ∗ z); + x ∗ insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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 dHochschild