Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

2019/08/12 by Mouquin, Victor
#Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1908.04044

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.

Related

• Double Bruhat cells and symplectic groupoids
• Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
• ON POISSON GROUPOIDS
• Morita equivalences, moduli spaces and flag varieties
• On Dynamical Poisson Groupoids I