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

Abstract

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 permutation group $\mathcal{G}(X,r)$. It induces a subgroup $\overline{\mathcal{G}}\subseteq \mathrm{Aut}(\mathcal{A}(X,r))$, the induced automorphism group, for which $\mathcal{A}(X,r)$ is a faithful module. We characterize when $\overline{\mathcal{G}}$ is a reflection group and, in that case, describe the invariant subalgebra $\mathcal{A}(X,r{)}^{\overline{\mathcal{G}}}$ together with its Jacobian, reflection arrangement and discriminant. We further establish the Auslander theorem for a large class of distributive Yang-Baxter algebras. Finally, for a class of nontrivial distributive Yang--Baxter algebras, we obtain a lower bound for the pertinency of the group action induced by the permutation group.