2020/03/17 by Chekhov, L., Shapiro, M.
#53D30 #81R12 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2003.07499
Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the \mathcal 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 \mathcal A3 and \mathcal A4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for \mathcal An via sequences of cluster mutations in the special \mathbb An-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.