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SUMMARY:Leonid Chekhov (Michigan State University and Steklov Mathematical
  Institute)
DTSTART:20210319T010000Z
DTEND:20210319T020000Z
DTSTAMP:20260423T004638Z
UID:CGP-MP/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CGP-MP/2/">D
 arboux coordinates for symplectic groupoid and cluster algebras</a>\nby Le
 onid Chekhov (Michigan State University and Steklov Mathematical Institute
 ) as part of IBS-CGP Mathematical Physics Seminar\n\n\nAbstract\nThe talk 
 is based on Arxiv:2003:07499\, joint work with Misha Shapiro. I will start
  with a short elementary excursion into cluster algebras---a fascinating b
 ranch of modern algebra introduced by Fomin and Zelvinsky in 2000's---and 
 describe  planar directed networks. We then concentrate on another interes
 ting algebraic object---the $\\mathcal A_n$ groupoid of upper-triangular m
 atrices\, which has had many appearances in studies of algebras of monodro
 mies of $SL_2$ Fuchsian systems and in geometry\, including the celebrated
  Goldman bracket. I will show how we can use Fock--Goncharov higher Teichm
 \\"uller space variables to derive  canonical (Darboux) coordinate represe
 ntation for entries of general symplectic leaves of the $\\mathcal A_n$ gr
 oupoid and\, in a more general setting\, of higher-dimensional symplectic 
 leaves for algebras governed by the quantum reflection equation with the t
 rigonometric $R$-matrix. For the groupoid of upper-triangular matrices\, w
 e represent braid-group transformations via sequences of cluster mutations
  in the special $\\mathbb A_n$-quiver. Time permitting\, I will also descr
 ibe a generalization of this construction to affine Lie-Poisson algebras a
 nd to quantum loop algebras (Arxiv:2012:10982).\n
LOCATION:https://researchseminars.org/talk/CGP-MP/2/
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