Infinitesimal Bialgebra on Planar Binary Trees

Abstract

We construct a weight-zero infinitesimal bialgebra structure on the $\bk$-module spanned by planar binary trees, using the under product $\backslash$ of Aguiar--Sottileand a root-recursive coproduct $\DeltaLR$. We prove that $\DeltaLR$ is coassociative and satisfies the infinitesimal derivation rule with respect to $\backslash$, hence gives a unitary infinitesimal bialgebra distinct from the usual Loday--Ronco Hopf structure. We also obtain an elementary vertex-cut formula, establish freeness properties for unitary $(\backslash ,\vee )$-algebras and unitary infinitesimal $(\backslash ,\vee )$-bialgebras, and identify the construction with the infinitesimal coproduct transported from planar rooted forests.

21 pages