Algèbres pré-Gerstenhaber à homotopie près

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.
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Submitted on : Tuesday, June 19, 2012 - 7:37:18 PM
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  • HAL Id : hal-00709757, version 1
  • ARXIV : 1206.4335

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Walid Aloulou, Didier Arnal, Ridha Chatbouri. Algèbres pré-Gerstenhaber à homotopie près. 2012. ⟨hal-00709757⟩

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