The infinitesimal Hopf algebra and the operads of planar forests

TL;DR

This paper introduces operads based on planar forests and explores their algebraic structures, including compatibility with coproducts and applications to dual basis computation, advancing the understanding of infinitesimal Hopf algebras.

Contribution

It defines new operads for planar forests, studies their algebraic compatibility, and provides inductive methods and theorems for infinitesimal Hopf algebra structures.

Findings

Operads with planar forests basis are introduced.

Compatibility with infinitesimal coproducts is established.

Inductive computation of dual bases is developed.

Abstract

We introduce two operads which own the set of planar forests as a basis. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees H becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of H and these structures is studied. As an application, an inductive way of computing the dual basis of H for its infinitesimal pairing is given. Moreover, three Cartier-Quillen-Milnor-Moore theorems are given for the operads of planar forests and a rigidity theorem for one of them.