Cohomology and extensions of relative Rota–Baxter groups

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.