Cluster structures on simple complex Lie groups and the Belavin-Drinfeld classification

TL;DR

This paper explores the relationship between cluster structures and Poisson-Lie structures on simple complex Lie groups, proposing a conjecture linking Belavin-Drinfeld classes to cluster structures and proving it for specific cases.

Contribution

It introduces a conjecture connecting Belavin-Drinfeld classification to cluster structures on Lie groups and proves it for certain groups and structures.

Findings

Established the conjecture for SL_n, n<5.

Proved the conjecture for standard Poisson-Lie structures.

Provided a reduction theorem linking different parts of the conjecture.

Abstract

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds to a cluster structure in $\O(\G)$. We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for $SL_n$, $n<5$, and for any $\G$ in the case of the standard Poisson-Lie structure.