Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests
Pierre J. Clavier, Douglas Modesto
TL;DR
This work extends shuffle algebra concepts from words to rooted trees and forests by constructing a shuffle bialgebra structure on rooted trees via a coproduct $\Delta$, compatible with a forest shuffle $\shuffle^T_\lambda$. It then develops a dual theory with a recursive dual coproduct $\Delta^*$ described combinatorially through admissible vertex families, and introduces pre-Lie grafting products that realize the dual bialgebra, establishing a canonical duality to the shuffle bialgebra. The primitive trees for the dual coproduct are characterized (internal vertices must have fertility at least two) and enumerated via a recursive formula, yielding concrete counts. Finally, the paper connects these combinatorial constructions to Rota-Baxter algebras by defining modified forest diamond products $\diamond_\lambda$ and proving a universal extension property, thereby linking rooted-tree combinatorics with pre-Lie, Grossman-Larson, and Rota-Baxter theories with potential implications for multiple zeta values, renormalization, and SPDEs.
Abstract
We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then caracterize the coproduct dual to the shuffle product of rooted forests and build a product on rooted trees to obtain the bialgebra dual to the shuffle bialgebra. We then characterize and enumerate primitive trees for the dual coproduct. Finally, using modified shuffles of rooted forests, we prove a property in the category of Rota-Baxter algebras.