Operated groups, differential groups and Rota-Baxter groups with an emphasis on the free objects

Abstract

Abstract–Groups with various types of operators, in particular the recently introduced Rota-Baxter groups, have generated renowned interest with close connections to numerical integrals, Yang-Baxter equation, integrable systems and post-Hopf algebras. This paper gives the general notion of operated groups and provides explicit constructions of free operated groups, free differential groups and free Rota-Baxter groups.

Keywords:

• Crossed homomorphism
• differential groups
• free object
• operated groups
• Rota-Baxter group

2020 Mathematics Subject Classification:

• 22E60
• 20E05
• 08B20
• 20D45
• 17B38
• 17B40
• 05E16

Data availability statement

The manuscript has no associated data.

Disclosure statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Acknowledgments

The authors thank the referees for their helpful reports.

Additional information

Funding

This work is supported by NSFC (12071191 and 12171211), the Natural Science Foundation of Jiangxi Province (20192ACB21008) and the Natural Science Foundation of Shandong Province (ZR2020MA002).