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 as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to "twist" a direct product of Poisson groupoids.

Cluster Structures on Double Bott-Samelson Cells

“…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].…”