Darboux coordinates for symplectic groupoid and cluster algebras
Abstract
Using FockGoncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of uppertriangular matrices and, in a more general setting, of higherdimensional 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 braidgroup 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.07499
 Bibcode:
 2020arXiv200307499C
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 81R12;
 53D30
 EPrint:
 41 pages, 33 figures, many corrections