Quantization of Poisson-Hopf stacks associated with group Lie bialgebras

Abstract

Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $\Gamma$ on $G$ deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting $\Gamma$ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a $\Gamma$ Lie bialgebra which is the infinitesimal of a $\Gamma$ Poisson-Lie group (cf \cite{MS} for simple Lie groups and $\Gamma$ a covering of the Weyl group and \cite{EH} for quantization in the general case).