Alg\`ebres de greffes

TL;DR

This paper constructs and analyzes a hierarchy of Hopf algebras of rooted trees using grafting operators, revealing their cofree, self-dual structure and generating properties within bigraft algebras.

Contribution

It introduces new grafting operators to build a family of Hopf algebras of rooted trees and studies their algebraic properties and structures.

Findings

The Hopf algebras form an increasing hierarchy with specific inclusion relations.

The algebra  is cofree and self-dual.

 is generated as a bigraft algebra by a single element.

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} $.