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 and apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair
r,r′. This is joint work with M. Shapiro and A. Vainshtein.
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