Relative Rota–Baxter groups and skew left braces

Abstract

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.