Vol. 317, No. 2, 2022

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Hopf algebra of multidecorated rooted forests, free matching Rota–Baxter algebras and Gröbner–Shirshov bases

Yi Zhang, Xing Gao and Li Guo

Vol. 317 (2022), No. 2, 441–475
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 Gröbner–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.

Keywords
rooted tree, Hopf algebra, Rota–Baxter algebra, 1-cocycle condition, Gröbner–Shirshov basis
Mathematical Subject Classification
Primary: 05E16, 16T30, 16Z10, 17B38
Secondary: 05A05, 05C05, 16S10, 16T10, 16W99
Milestones
Received: 30 June 2021
Revised: 20 December 2021
Accepted: 20 January 2022
Published: 14 July 2022
Authors
Yi Zhang
School of Mathematics and Statistics, Nanjing University of Information Science & Technology
Nanjing
China
Xing Gao
School of Mathematics and Statistics
Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University
Lanzhou
China
Li Guo
Department of Mathematics and Computer Science
Rutgers University
Newark, NJ
USA