Extended Goldman symplectic structure in Fock–Goncharov coordinates

Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant under natural transformations. We use this notion to identify the canonical non-degenerate extension of Goldman's symplectic form on the $SL(n)$ character variety with a the form associated to a suitable graph. Using the invariance of the form under natural moves, we utilize the parametrization of the character variety in terms of Fock--Goncharov coordinates and associate to it a canonical decorated triangulation. This allows us to show that these coordinates are log--canonical for the extended Goldman Poisson structure.