2019
DOI: 10.48550/arxiv.1908.04044
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Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Abstract: Given a standard complex semisimple Poisson Lie group (G, π st ), generalised double Bruhat cells G u,v and generalised Bruhat cells O u equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang-Hua Lu and the author. We prove in this paper that G u,u is naturally a Poisson groupoid over O u , extending a result from the aforementioned authors about double Bruhat cells in (G, π st ).Our result on G u,u is obtained a… Show more

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“…In a recent work, Mouquin [Mou19] proved that the generalized double Bruhat cell G u,u {e} is a Poisson groupoid over the generalized Bruhat cell G e,u {e} . We believe that the Poisson groupoid structure coincides with Fock-Goncharov's symplectic double for cluster varieties [FG09b].…”
Section: Further Questionsmentioning
confidence: 99%
“…In a recent work, Mouquin [Mou19] proved that the generalized double Bruhat cell G u,u {e} is a Poisson groupoid over the generalized Bruhat cell G e,u {e} . We believe that the Poisson groupoid structure coincides with Fock-Goncharov's symplectic double for cluster varieties [FG09b].…”
Section: Further Questionsmentioning
confidence: 99%