Pre-Gerstenhaber Algebra to nearest homotopy

This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations. and [,]. An important example of such a structure is the Gerstenhaber algebra (i.e. commutatitve structure with degree 0 and Lie structure with degree -1). The notion of Gerstenhaber algebra up to homotopy (G(infinity), algebra) is known: it is a codifferential bicogebra. Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre Lie) allowing a similar construction for a preG(infinity) algebra. Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding Koszul dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy: it is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts. (C) 2017 Elsevier B.V. Tous droits reserves.