Alg\`ebres de greffes

Abstract

In order to study some sets of probabilities, called induced averages by J. Ecalle, F. Menous introduces two grafting operators $ B^{+} $ and $ B^{-} $. With these two operators, we construct Hopf algebras of rooted and ordered trees $ \mathcal{B}^{i} $, $ i \in \mathbb{N}^{\ast} $, $ \mathcal{B}^{\infty} $ and $ \mathcal{B} $ satisfying the inclusion relations $ \mathcal{B}^{1} \subseteq \hdots \mathcal{B}^{i} \subseteq \mathcal{B}^{i+1} \subseteq \hdots \subseteq \mathcal{B}^{\infty} \subseteq \mathcal{B} $. We endow $ \mathcal{B} $ with a structure of duplicial dendriform bialgebra and we deduce that $ \mathcal{B} $ is cofree and self-dual. Finally, we introduce the notion of bigraft algebra and we prove that $ \mathcal{B} $ is generated as bigraft algebra by the element $ \tdun{1} $.