Log-Canonical Coordinates for Symplectic Groupoid and Cluster Algebras

Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of the Aₙ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A₃ and A₄ in terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations for Aₙ via sequences of cluster mutations in the special Aₙ-quiver. We prove the groupoid relations for normalized quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network.Dedicated to the memory of a great mathematician and person, Boris Dubrovin.