Species of Rota-Baxter algebras by rooted trees, twisted bialgebras and Fock functors

TL;DR

This paper explores the combinatorial and algebraic structures of Rota-Baxter algebras using species, rooted trees, and twisted bialgebras, providing new insights into their free constructions and bialgebra properties.

Contribution

It introduces a species of Rota-Baxter algebras based on decorated forests, proves its freeness, and establishes a twisted bialgebra structure using combinatorial and Fock functor methods.

Findings

The species of Rota-Baxter algebras is shown to be free.

A twisted bialgebra structure is constructed on this species.

Fock functor provides an alternative proof of the bialgebra structure.

Abstract

As a fundamental and ubiquitous combinatorial notion, species has attracted sustained interest, generalizing from set-theoretical combinatorial to algebraic combinatorial and beyond. The Rota-Baxter algebra is one of the algebraic structures with broad applications from Renormalization of quantum field theory to integrable systems and multiple zeta values. Its interpretation in terms of monoidal categories has also recently appeared. This paper studies species of Rota-Baxter algebras, making use of the combinatorial construction of free Rota-Baxter algebras in terms of angularly decorated trees and forests. The notion of simple angularly decorated forests is introduced for this purpose and the resulting Rota-Baxter species is shown to be free. Furthermore, a twisted bialgebra structure, as the bialgebra for species, is established on this free Rota-Baxter species. Finally, through the Fock functor, another proof of the bialgebra structure on free Rota-Baxter algebras is obtained.