QUANTIZATION OF POISSON-HOPF STACKS ASSOCIATED WITH GROUP LIE BIALGEBRAS

Let G be a simply connected 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. This induces the existence of a Poisson-Hopf algebra structure on the direct sum over Gamma of formal functions on G, with Poisson structures translated by Gamma. A quantization of this algebra can be obtained by taking the linear dual of a quantization of the Gamma Lie bialgebra g, which is the infinitesimal of a Gamma Poisson-Lie group. In this paper we find out an interesting structure on the dual Lie group G*. We prove that we can construct a stack of Poisson-Hopf algebras and prove the existence of the associated deformation quantization of it. This stack can be viewed as the function algebra on "the formal Poisson group" dual to the original Gamma Poisson-Lie group. To quantize this stack, we apply Drinfeld functors to quantization of the associated Gamma Lie bialgebra.