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

TL;DR

This paper proves a conjecture linking cluster structures and Poisson brackets on SL_n, showing a one-to-one correspondence with Belavin-Drinfeld data and establishing an isomorphism between the upper cluster algebra and the coordinate ring.

Contribution

It completes the proof of a conjecture relating cluster structures to Belavin-Drinfeld data on SL_n, establishing a precise correspondence and algebraic isomorphism.

Findings

The upper cluster algebra is isomorphic to the coordinate ring of SL_n.

The torus from the BD triple acts on the coordinate ring.

There is a one-to-one correspondence between BD classes and cluster structures.

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.