Summary: STRUCTURE OF THE LODAY-RONCO
HOPF ALGEBRA OF TREES
MARCELO AGUIAR AND FRANK SOTTILE
Abstract. Loday and Ronco defined an interesting Hopf algebra structure on the linear
span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra
of non-commutative symmetric functions in the Malvenuto-Reutenauer Hopf algebra of per-
mutations factors through their Hopf algebra of trees, and these maps correspond to natural
maps from the weak order on the symmetric group to the Tamari order on planar binary
trees to the boolean algebra.
We further study the structure of this Hopf algebra of trees using a new basis for it.
We describe the product, coproduct, and antipode in terms of this basis and use these
results to elucidate its Hopf-algebraic structure. We also obtain a transparent proof of its
isomorphism with the non-commutative Connes-Kreimer Hopf algebra of Foissy, and show
that this algebra is related to non-commutative symmetric functions as the (commutative)
Connes-Kreimer Hopf algebra is related to symmetric functions.
1. Basic Definitions 2
2. Some Galois connections between posets 10
3. Some Hopf morphisms involving YSym 13
4. Geometric interpretation of the product of YSym 14