Yang-Baxter permutation group actions on distributive Yang-Baxter algebras

Let $(X,r)$ be a distributive set-theoretical solution of the Yang-Baxter equation and $\mathcal{A}(X,r)$ the associated Yang-Baxter algebra. We prove that $\mathcal{A}(X,r)$ is isomorphic to a skew polynomial algebra and compute its Nakayama automorphism explicitly. We study the action of the permu