The Infinitesimal Hopf Algebra and the Operads of Planar Forests

We introduce two operads based on the set of planar forests. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees $\mathcal H$ becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of $\mathcal H$ and these structures is studied. As an application, an inductive way of computing the dual basis of $\mathcal H$ 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.