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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