Exotic Cluster Structures on $SL_n$ with Belavin-Drinfeld Data of Minimal Size, I. The Structure

TL;DR

This paper constructs and verifies a cluster structure on SL_n associated with minimal Belavin-Drinfeld data, establishing compatibility with the Poisson brackets and laying groundwork for a conjectured correspondence.

Contribution

It provides an explicit algorithm for constructing initial seeds for cluster structures on SL_n linked to minimal Belavin-Drinfeld data and proves their compatibility with Poisson brackets.

Findings

Constructed initial seeds for cluster structures on SL_n.

Proved compatibility of cluster structures with Poisson brackets.

Established local regularity of the seeds.

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 $SL_{n}$, we give an algorithm for constructing an initial seed $\Sigma$ in $\mathcal{O}(SL_{n})$. The cluster structure $\mathcal{C}=\mathcal{C}(\Sigma)$ is then proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data, and the seed $\Sigma$ is locally regular. This is the first of two papers, and the second one proves the rest of the conjecture: the upper cluster algebra $\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}(SL_{n})$, and the correspondence of Belavin-Drinfeld classes and cluster structures is one to one.