A Global Quantum duality Principle for Subgroups and Homogeneous Spaces

For a complex or real algebraic group G, with g := Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{−1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g: correspondingly, one has dual Poisson groups G^∗ and a dual Lie bialgebra g^∗. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called flat in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever. The global quantum duality principle (GQDP), as developed in [F. Gavarini, The global quantum duality principle, Journ. fur die Reine Angew. Math. 612 (2007), 17–33.], associates with any global quantization of G, or of g, a global quantization of g^∗, or of G^∗. In this paper we present a similar GQDP for quantum subgroups or quantum homogeneous spaces. Roughly speaking, this associates with every quantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneous space, resp. a quantum subgroup, of G^∗. The construction is tailored after four parallel paths — according to the different ways one has to algebraically describe a subgroup or a homogeneous space — and is “functorial”, in a natural sense. Remarkably enough, the output of the constructions are always quantizations of proper type. More precisely, the output is related to the input as follows: the former is the coisotropic dual of the coisotropic interior of the latter — a fact that extends the occurrence of Poisson duality in the original GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict as well — so the special role of strict quantizations is respected. We end the paper with some explicit examples of application of our recipes.