Exotic cluster structures on $SL_n$ with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples

Abstract: Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin--Drinfeld classification of solutions to the classical Yang Baxter equation. For any non trivial Belavin--Drinfeld data of minimal size for $SL_{n}$, the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin--Drinfeld data. This paper proves the rest of the conjecture: the corresponding upper cluster algebra $\overline{\mathcal{A}}{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}\left(SL{n}\right)$, the torus determined by the BD triple generates theaction of $(\mathbb{C}^{*})^{2k_{T}}$ on $\mathbb{C}\left(SL_{n}\right)$, and the correspondence between Belavin--Drinfeld classes and cluster structures is one to one.