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 Fq[G] or Uq(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. f̈ur 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 Documenta Mathematica 19 (2014) 333-380 334 Nicola Ciccoli, Fabio Gavarini input is a strict quantization then the output is strict as well - so the special r̂ ole of strict quantizations is respected. We end the paper with some explicit examples of application of our recipes.