|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
|
