Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gr\"obner-Shirshov bases

Abstract

Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gr\"obner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxter algebras are equipped with a cocycle Hopf algebra structure.