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arXiv · 2606.04372

Infinitesimal Bialgebra on Planar Binary Trees

Yong Yu·Xing Gao
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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.

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Yong Yu, Xing Gao. 2026-06-03. Infinitesimal Bialgebra on Planar Binary Trees. https://arxiv.org/abs/2606.04372

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