Cohomology and Extensions of Relative Rota-Baxter groups

TL;DR

This paper develops an extension theory and cohomology framework for relative Rota-Baxter groups, revealing their connections to skew left braces and the Yang-Baxter equation, and establishing a cohomological classification of extensions.

Contribution

It introduces a new low-dimensional cohomology theory for relative Rota-Baxter groups and links it to existing cohomologies of associated algebraic structures.

Findings

Established a bijection between extension classes and second cohomology.

Connected the cohomology of relative Rota-Baxter groups with that of skew left braces.

Proved isomorphism of cohomologies in the bijective case.

Abstract

Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation. In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two.