Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach

TL;DR

This paper develops a new algebraic framework for decorated rooted forests, introducing coproducts, dual structures, and cocycle conditions, and explores their universal properties within operated algebra theory.

Contribution

It provides a novel operated algebra approach to Moerdijk Hopf algebras, including explicit coproduct formulations and the concept of forest-representable matrices.

Findings

Constructed a bialgebra and Hopf algebra structure on decorated forests.

Introduced the notion of forest-representable matrices for dual coproducts.

Connected the antipode with Rota-Baxter operators and established universal properties.

Abstract

In this paper, we first endow the space of decorated planar rooted forests with a coproduct that equips it with the structure of a bialgebra and further a Moerdijk Hopf algebra. We also present a combinatorial description of this coproduct, and further give an explicit formulation of its dual coproducts through the newly defined notion of forest-representable matrices. By viewing the Moerdijk Hopf algebra within the framework of operated algebras, we introduce the notion of a multiple cocycle Hopf algebra, incorporating a symmetric Hochschild 1-cocycle condition. We then show that the antipode of this Hopf algebra is a Rota-Baxter operator on Moerdijk Hopf algebras. Furthermore, we investigate the universal properties of cocycle Hopf algebras. As an application, we construct the initial object in the category of free cocycle Hopf algebras on undecorated planar rooted forests, which coincides with the well-known Moerdijk Hopf algebra.