A quantum duality principle for subgroups and homogeneous spaces

Abstract

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any quantum subgroup (in a tautological sense) of the initial Poisson group, while in the second one similar recipes are given only starting from coisotropic subgroups. In both cases this yields a Galois-type correspondence, where a quantum coisotropic subgroup is mapped to its complementary dual; moreover, in the first formulation quantum coisotropic subgroups are characterized as being the fixed points in this Galois' reciprocity. By the natural link between quantum subgroups and quantum homogeneous spaces then we argue a quantum duality principle for homogeneous spaces too, where quantum coisotropic spaces are the fixed elements in a suitable Galois' reciprocity. As an application, we provide an explicit quantization of the space of Stokes matrices with the Poisson structure given by Dubrovin and Ugaglia. The *paper is actually under revision* to fix some minor, technical issues. The geometric objects considered here (Poisson groups, subgroups and homogeneous spaces) are "global" - as opposed to "formal" - and quantizations are considered as standard - i.e.non topological - Hopf algebras over the ring of Laurent polynomials (or other non-topological rings). Instead, a "local/formal" version of this work is developed in math.QA/0412465 - which is in final form - due to appear in "Advances in Mathematics". The example of Stokes matrices mentioned above is considered in math.QA/0412465 as well.