Extensions and automorphisms of Rota-Baxter groups

TL;DR

This paper develops a cohomology theory for Rota-Baxter groups, explores their extensions, and generalizes the Wells automorphism sequence, advancing the algebraic understanding of these structures.

Contribution

It introduces a cohomology framework for Rota-Baxter groups and connects their extensions to those of skew braces, also generalizing automorphism sequences.

Findings

Constructed a cohomology theory for Rota-Baxter groups

Established relations between Rota-Baxter group extensions and skew brace extensions

Generalized the Wells short exact sequence for automorphisms

Abstract

The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [{\em Adv. Math.} 387 (2021), 107834, 34 pp.] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter groups by constructing suitable cohomology theories. Among others, we find relations with the extensions of skew braces. Given an extension of Rota-Baxter groups, we also construct a short exact sequence connecting various automorphism groups, which generalizes the Wells short exact sequence.