BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Leonid Chekhov (Steklov Mathematical Institute and Michigan State University) DTSTART:20210202T160000Z DTEND:20210202T170000Z DTSTAMP:20260423T021134Z UID:OCAS/18 DESCRIPTION:Title: Da rboux coordinates for symplectic groupoid and cluster algebras\nby Leo nid Chekhov (Steklov Mathematical Institute and Michigan State University) as part of Online Cluster Algebra Seminar (OCAS)\n\n\nAbstract\nThe talk is based on Arxiv:2003:07499\, joint work with Misha Shapiro. We use Fock- -Goncharov higher Teichmüller space variables to derive Darboux coordina te representation for entries of general symplectic leaves of the $\\mathc al A_n$ groupoid of upper-triangular matrices and\, in a more general sett ing\, of higher-dimensional symplectic leaves for algebras governed by the quantum reflection equation with the trigonometric $R$-matrix. This resul t can be generalized to any planar directed network on disc with separated sinks and sources. For the groupoid of upper-triangular matrices\, we rep resent braid-group transformations via sequences of cluster mutations in t he special $\\mathbb A_n$-quiver. We prove the groupoid relations for quan tum transport matrices and\, as a byproduct\, obtain quantum commutation r elations having the Goldman bracket as their semiclassical limit. Time per mitting\, I will also describe a generalization of this construction to af fine Lie-Poisson algebras and to quantum loop algebras (Arxiv:2012:10982). \n LOCATION:https://researchseminars.org/talk/OCAS/18/ END:VEVENT END:VCALENDAR