Orbit Finiteness in Skew Braces

This talk is based on joint work with R. Cascella and A. Van Antwerpen. We investigate structural properties of skew braces and their connection with set-theoretic solutions of the Yang–Baxter equation, highlighting how group-theoretic ideas extend to this setting, while also revealing more subtle and less rigid behaviour. Our approach is based on natural actions, λ and θ, arising from the internal structure of a skew brace and from the theory of set-theoretic solutions of the Yang–Baxter equation. In the group-theoretic setting, these constructions are analogous to those that arise from conjugation and group actions. The resulting orbit decompositions lead to the definition of the FC-type conditionsλf and θ_f. These finiteness conditions arise naturally in the study of finite set-theoretic solutions of the Yang–Baxter equation. Indeed, the structure skew brace of every such solution is λ_f, θ_f, and its additive group is FC. These conditions exhibit several properties analogous to finite conjugacy in group theory, including the existence of an analogue of the FC-centre and related structural results. Furthermore, when considering the structure skew brace of a solution, these properties measure whether elements are contained in a finite decomposition factor, identifying a class of infinite solutions that may exhibit properties similar to finite ones.

Sep 3, 2026