Factorizations of skew braces

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang–Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô’s theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang–Baxter equation. Finally, we classify skew braces that contain no non-trivial proper characteristic ideals.Fil: Jespers, E.. Vrije Unviversiteit Brussel; BélgicaFil: Kubat, L.. Vrije Unviversiteit Brussel; BélgicaFil: Van Antwerpen, A.. Vrije Unviversiteit Brussel; BélgicaFil: Vendramin, Claudio Leandro. Institute of Mathematical Sciences at NYU Shanghai; China. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

On solubility of skew left braces and solutions of the Yang-Baxter equation

The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, we describe finite skew left braces admitting no proper substructure, we introduce a suitable notion of solubility of skew left braces and study the ideal structure of soluble skew left braces. As a consequence, results on decomposability of solutions are also obtained.Comment: 18 page

Some group-theoretical approaches to skew left braces

The algebraic structure of skew left brace has become a useful tool to construct set-theoretic solutions of the Yang-Baxter equation. In this survey we present some descriptions of skew left braces in terms of bijective derivations, triply factorised groups, and regular subgroups of the holomorph of a group, as well as some applications of these descriptions to the study of substructures, nilpotency, and factorised skew left braces

Isoclinism of skew braces

We define isoclinism of skew braces and present several applications. We study some properties of skew braces that are invariant under isoclinism. For example, we prove that right nilpotency is an isoclinism invariant. This result has application in the theory of set-theoretic solutions to the Yang-Baxter equation. We define isoclinic solutions and study multipermutation solutions under isoclinism.Comment: 14 pages. Postprint versio

Cohomology and Extensions of Relative Rota-Baxter groups

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.Comment: 30 page

Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.Comment: 18 pages. Postprint versio