Кластерные координаты Дарбу для симплектического группоида

Using Fock–Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ 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. We represent braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We also obtain quantum algebras of transport paths for any directed planar network—acyclic or cyclic. Time permitting, I will also present recent results on constructing realizations of affine generalizations of Lie–Poisson and reflection equations based on quantum algebras of transport paths.