A global quantum duality principle for subgroups and homogeneous spaces

Abstract

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 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. The global quantum duality principle (GQDP) - cf. [F. Gavarini, "The global quantum duality principle", J. 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 GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict too - so the special role of strict quantizations is respected. We end the paper with some examples and application.