Exotic cluster structures on SL _n with Belavin–Drinfeld data of minimal size, II. Correspondence between cluster structures and Belavin–Drinfeld 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 the existence of a cluster structure for each Belavin-Drinfeld solution of the classical Yang-Baxter equation compatible with the corresponding Poisson-Lie bracket on the simple Lie group. 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 $\overline {{A_\mathbb{C}}} \left( C \right)$ is naturally isomorphie to O (SL _n ), the torus determined by the BD triple generates the action of ${\left( {\mathbb{C}*} \right)^{2{k_T}}}$ on C (SL _n ), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.