Species of Rota-Baxter algebras by rooted trees, twisted bialgebras and Fock functors
Loic Foissy, Li Guo, Xiao-Song Peng, Yunzhou Xie, Yi Zhang
TL;DR
The paper develops a categorical framework linking combinatorial species with Rota-Baxter algebras by decorating rooted forests. It constructs a free Rota-Baxter species on simple angularly decorated forests and endows it with a twisted bialgebra structure via universal properties, with additional RB-tensor power structures. The Fock functor (including colored variants) provides an alternative proof path and recovers the classical free Rota-Baxter algebras from tree-like combinatorics. Overall, the work deepens the interplay between species theory, RB-algebraic structures, and category-theoretic methods, with potential implications for renormalization, combinatorial Hopf-type algebras, and related areas in algebraic combinatorics.
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.