A global quantum duality principle for subgroups and homogeneous spaces

Ciccoli, Nicola ; Gavarini, Fabio

Documenta mathematica, Tome 19 (2014), pp. 333-380.

Résumé

Summary: 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.

Détail

Classification : 17B37, 20G42, 58B32, 81R50
Keywords: quantum groups, Poisson homogeneous spaces, coisotropic subgroups