#### DOCUMENTA MATHEMATICA, Vol. 19 (2014), 333-380

Nicola Ciccoli, Fabio Gavarini

A Global Quantum Duality Principle for Subgroups and Homogeneous Spaces

For a complex or real algebraic group $G$, with $\mathfrak{g} := \mathrm{Lie}(G)$, quantizations of {\sl global\/} type are suitable Hopf algebras $F_q[G]$ or $U_q(\mathfrak{g})$ over $\{C}\big[q,q^{-1}\big]$. Any such quantization yields a structure of Poisson group on $G$, and one of Lie bialgebra on $\mathfrak{g}$: correspondingly, one has dual Poisson groups $G^*$ and a dual Lie bialgebra $\mathfrak{g}^*$. In this context, we introduce suitable notions of {\sl quantum subgroup\/} and, correspondingly, of {\sl quantum homogeneous space}, in three versions: {\sl weak}, {\sl proper\/} and {\sl strict\/} (also called {\sl 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, {\it The global quantum duality principle}, Journ.für die Reine Angew.Math.{\bf 612} (2007), 17--33.], associates with any global quantization of $G$, or of $\mathfrak{g}$, a global quantization of $\mathfrak{g}^*$, 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 {\sl proper\/} type. More precisely, the output is related to the input as follows: the former is the {\it 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 rôle of strict quantizations is respected.

We end the paper with some explicit examples of application of our recipes.

2010 Mathematics Subject Classification: Primary 17B37, 20G42, 58B32; Secondary 81R50

Keywords and Phrases: Quantum Groups, Poisson Homogeneous Spaces, Coisotropic Subgroups

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