Relative Rota-Baxter groups and skew left braces

TL;DR

This paper establishes a deep connection between relative Rota-Baxter groups and skew left braces, showing they are essentially equivalent and providing computational tools and structural insights into their relationship.

Contribution

It proves the equivalence between relative Rota-Baxter groups and skew left braces, introduces an efficient GAP algorithm, and explores the concept of isoclinism in this context.

Findings

Every relative Rota-Baxter group yields a skew left brace.

Every skew left brace arises from a relative Rota-Baxter group.

An isoclinism of these groups induces an isoclinism of the associated braces.

Abstract

Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and 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 non-degenerate set-theoretic 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.