Alg\`ebres pr\'e-Gerstenhaber \`a homotopie pr\`es

TL;DR

This paper introduces the concept of pre-Gerstenhaber algebras up to homotopy, providing explicit constructions and extending the framework of Gerstenhaber algebras to a pre-structure setting.

Contribution

It defines pre-Gerstenhaber algebras up to homotopy and constructs explicit models using dual operads and bicogebras, extending known homotopy algebra concepts.

Findings

Defined pre-Gerstenhaber algebra up to homotopy.

Constructed explicit models using dual operads.

Extended homotopy algebra frameworks to pre-structures.

Abstract

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 (commutatitve and Lie). The notion of Gerstenhaber algebra up to homotopy ($G_\infty$ algebra) is known. Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing the construction of $\hbox{pre}G_\infty$ algebra. Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy, this is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts.