Braided dendriform and tridendriform algebras, and braided Hopf algebras of rooted trees

Abstract

This paper studies the braidings of several Hopf algebras of rooted trees which have found broad applications. First by studying free braided dendriform algebras, we obtain the braiding of the Loday–Ronco Hopf algebra of planar binary rooted trees. We also give a variation of the braiding obtained by Foissy for the noncommutative Connes–Kreimer (a.k.a the Foissy–Holtkamp) Hopf algebra of planar rooted forests, and show that the well-known isomorphism between this Hopf algebra and the Loday–Ronco Hopf algebra can be extended to the braided context. We further study braided tridendriform algebras and apply their free objects to give a braiding of the Hopf algebra of Loday and Ronco on planar angularly decorated trees. Along the way, we obtain a combinatorial characterization and a generating function for such trees.