Infinitesimal Bialgebra on Planar Binary Trees

Abstract

We construct a weight-zero infinitesimal bialgebra structure on the $\mathbf{k}$-module spanned by planar binary trees, using the under product $\backslash$ of Aguiar-Sottile and a root-recursive coproduct $\Delta_{\rm LR}^{\epsilon}$. We prove that $\Delta_{\rm LR}^{\epsilon}$ 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 cut formula, establish freeness properties for unitary $(\backslash,\vee)$-algebras and unitary infinitesimal $(\backslash,\vee)$-bialgebras, and build an infinitesimal bialgebra isomorphism from the infinitesimal bialgebra of planar binary trees to the known infinitesimal bialgebra of planar rooted forests.