Double Bruhat cells and symplectic groupoids (Q1757167)

In this paper, the authors construct a pair of dual action Poisson groupoids associated to quasitriangular \(r\)-matrices. The paper starts with some recalls about Poisson Lie groups and Lie bialgebras. Some properties of the standard complex semisimple Poisson Lie groups are also reviewed and proved. Then the authors prove that for a connected complex semidimple Lie group, and for each \(v\) in the Weyl group of \(G\), the corresponding double Bruhat cell with the Poisson structure, is a Poisson groupoid. This work generalizes also some results of Kogan and Zelevinsky.