A unified approach to exotic cluster structures on simple Lie groups

TL;DR

This paper introduces a unified method for constructing log-canonical coordinate charts on simple Lie groups with various Poisson structures, extending cluster algebra techniques to a broad class of Lie groups.

Contribution

It develops a new approach to build coordinate charts for any simple Lie group with arbitrary Poisson brackets linked to Belavin--Drinfeld data, including invertibility and cluster structure results.

Findings

Constructed rational maps between different Poisson structures on Lie groups.

Proved invertibility of maps in the $A_n$ case for aperiodic Belavin--Drinfeld data.

Established the existence of compatible regular complete cluster structures.

Abstract

We propose a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group $\G$ and arbitrary Poisson-homogeneous bracket on $\G$ associated with Belavin--Drinfeld data. Given a pair of representatives $r, r'$ from two arbitrary Belavin--Drinfeld classes, we build a rational map from $\G$ with the Poisson structure defined by two appropriately selected representatives from the standard class to $\G$ equipped with the Poisson structure defined by the pair $r, r'$. In the $A_n$ case, we prove that this map is invertible whenever the pair $r, r'$ is drawn from aperiodic Belavin--Drinfeld data, as defined in~\cite{GSVple}. We further apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair $r, r'$ in the aperiodic case.