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

TL;DR

This paper introduces the general concept of operated groups, focusing on explicit constructions of free operated, differential, and Rota-Baxter groups, highlighting their mathematical significance and connections to various areas.

Contribution

It provides the first explicit constructions of free operated groups, including differential and Rota-Baxter groups, advancing the understanding of their algebraic structures.

Findings

Explicit constructions of free operated groups

Development of free differential groups

Construction of free Rota-Baxter groups

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.