Darboux coordinates for symplectic groupoid and cluster algebras

Abstract

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. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for ${\mathcal{A}}_3$ and ${\mathcal{A}}_4$ in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for ${\mathcal{A}}_n$ via sequences of cluster mutations in the special ${\mathbb{A}}_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.

41 pages, 33 figures, many corrections