Hopf algebra structure on free Rota-Baxter algebras by angularly decorated rooted trees

TL;DR

This paper introduces a combinatorial approach to construct a Hopf algebra structure on free Rota-Baxter algebras using angularly decorated rooted forests, expanding the algebraic framework with new coproducts.

Contribution

It provides a novel combinatorial construction of a Hopf algebra on free Rota-Baxter algebras via angularly decorated rooted forests, which was not previously established.

Findings

Established a coproduct on free Rota-Baxter algebras using decorated rooted forests

Proved the resulting algebra has a bialgebra and Hopf algebra structure

Developed a new notion of subforests for combinatorial construction

Abstract

By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota-Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota-Baxter algebra with a bialgebra structure and further a Hopf algebra structure.