A unified approach to exotic cluster structures on simple Lie groups

We propose a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group $\text{G}$ and arbitrary Poisson-homogeneous bracket on $\text{G}$ associated with Belavin–Drinfeld data. Given a pair of representatives $r,r^{\mathrm{\prime }}$ from two arbitrary Belavin–Drinfeld classes, we build a rational map from $\text{G}$ with the Poisson structure defined by two appropriately selected representatives from the standard class to $\text{G}$ equipped with the Poisson structure defined by the pair $r,r^{\mathrm{\prime }}$. In the $A_n$ case, we prove that this map is invertible whenever the pair $r,r^{\mathrm{\prime }}$ 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 $r,r^{\mathrm{\prime }}$ in the aperiodic case.