The infinitesimal Hopf algebra and the operads of planar forests

Loïc Foissy (1)

1 LMR - Laboratoire de Mathématiques de Reims

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.

Mots clés: Operads; Infinitesimal Hopf algebra; Planar rooted trees; Operads.

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. The infinitesimal Hopf algebra and the operads of planar forests. International Mathematics Research Notices, 2010. ⟨hal-00353236⟩