Affine Rota-Baxter groups and affine skew braces

Abstract: Rota-Baxter groups and skew braces are closely related algebraic structures, both providing set-theoretical solutions to the Yang-Baxter equation. In this paper, we extend these structures to the setting of affine schemes. First, we introduce affine Rota-Baxter groups and, by leveraging the duality between affine groups and Hopf algebras via their coordinate rings, prove the equivalence between affine Rota-Baxter groups and co-Rota-Baxter Hopf algebras. Next, we show affine Rota-Baxter groups can naturally give rise to the affine skew braces defined by Angiono, Galindo, and Vendramin. Conversely, any affine skew brace can be embedded into an affine Rota-Baxter group. By linking these to the relationship between affine skew braces and Hopf co-braces, we give new connections between co-Rota-Baxter Hopf algebras and Hopf co-braces. Finally, we propose the study of solutions to the Yang-Baxter equation within the framework of affine schemes, demonstrating that affine skew braces naturally give rise to such solutions.