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.

30 pages

References in corpus (10)

• Integration and geometrization of Rota-Baxter Lie algebras
• Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
• Rota-Baxter operators on groups
• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Extensions and automorphisms of Rota-Baxter groups
• Relative Rota-Baxter groups and skew left braces
• Extensions and Well's type exact sequence of skew braces
• Schur multiplier and Schur covers of relative Rota-Baxter groups