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摘要
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 C, 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 C, 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.
- 出版日期2014
