Rota-Baxter operators on groups

TL;DR

This paper explores the concept of Rota-Baxter operators on groups, providing new constructions, examining extensions of maps to such operators, and analyzing their connections to Lie rings and specific simple groups.

Contribution

It introduces general methods for constructing Rota-Baxter operators on groups and investigates their relationships with Lie rings and sporadic simple groups.

Findings

Established new constructions of Rota-Baxter operators on groups.

Analyzed extensions of maps to Rota-Baxter operators.

Connected Rota-Baxter operators on groups to those on Lie rings.

Abstract

Theory of Rota-Baxter operators on rings and algebras has been developed since 1960. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] have defined the notion of Rota-Baxter operator on a group. We provide some general constructions of Rota-Baxter operators on a group. Given a map on a group, we study its extensions to a Rota-Baxter operator. We state the connection between Rota-Baxter operators on a group and Rota-Baxter operators on an associated Lie ring. We describe Rota-Baxter operators on sporadic simple groups.