A global quantum duality principle for subgroups and homogeneous spaces

Abstract

For a complex or real algebraic group $G$, with $\mathfrak{g}:=\mathrm{Lie}(G)$, quantizations of global type are suitable Hopf algebras $F_q[G]$ or $U_q(\mathfrak{g})$ over $C[q,q^{-1}]$. 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 $G^{\ast }$ and a dual Lie bialgebra ${\mathfrak{g}}^{\ast }$. 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.