For a complex or real algebraic group G, with g:=Lie(G), quantizations ofglobal type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Anysuch quantization yields a structure of Poisson group on G, and one of Liebialgebra on g : correspondingly, one has dual Poisson groups G^* and a dualLie bialgebra g^*. In this context, we introduce suitable notions of quantumsubgroup and of quantum homogeneous space, in three versions: weak, proper andstrict (also called "flat" in the literature). The last two notions only applyto those subgroups which are coisotropic, and those homogeneous spaces whichare Poisson quotients; the first one instead has no restrictions. The global quantum duality principle (GQDP) - cf. [F. Gavarini, "The globalquantum duality principle", J. Reine Angew. Math. 612 (2007), 17-33] -associates with any global quantization of G, or of g, a global quantization ofg^*, or of G^*. In this paper we present a similar GQDP for quantum subgroupsor quantum homogeneous spaces. Roughly speaking, this associates with everyquantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneousspace, resp. a quantum subgroup, of G^*. The construction is tailored afterfour parallel paths - according to the different ways one has to algebraicallydescribe a subgroup or a homogeneous space - and is "functorial", in a naturalsense. Remarkably enough, the output of the constructions are always quantizationsof 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 GQDP for quantumgroups. Finally, when the input is a strict quantization then the output isstrict too - so the special role of strict quantizations is respected. We end the paper with some examples and application.
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