The Poisson-Fourier Transform for bicrossed products I: Abelian approximations and the quantum duality principle

The quantum duality Principle of Drinfel’d states that any quantization ${\mathcal{G}}_{\mathrm{\hslash }}$ of a Poisson-Lie group $\mathcal{G}$ should be dual as a quantum group to a quantization ${\mathcal{G}}_{\mathrm{\hslash }}^{\ast }$ of the Poisson dual group ${\mathcal{G}}^{\ast }\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}$. In this paper we consider pairs $(\mathcal{G}=G⋉V,{\mathcal{G}}^{\ast }=H⋉W)$ with $V,W$ abelian, where we can realise the quantizations ${\mathcal{G}}_{\mathrm{\hslash }}$ and ${\mathcal{G}}_{\mathrm{\hslash }}^{\ast }$ as a bicrossed product between $G$ and $H$ in the setting of locally compact quantum groups. Assuming the existence of suitable maps ${\eta }_G:G\to \widehat{W}$ and ${\eta }_H:H\to \widehat{V}$ which we call abelian approximations, we implement the quantum duality principle by constructing an explicit unitary operator ${\mathcal{F}}_{\mathcal{G}}:{\mathrm{L}}^2(\mathcal{G})\to {\mathrm{L}}^2({\mathcal{G}}^{\ast })$, the Poisson-Fourier transform between $\mathcal{G}$ and ${\mathcal{G}}^{\ast }$. It induces an isomorphism of locally compact quantum group ${\mathcal{F}}_{\mathcal{G}}:{\widehat{\mathcal{G}}}_{\mathrm{\hslash }}\cong {\mathcal{G}}_{\mathrm{\hslash }}^{\ast }$. After discussing the general framework for the Poisson-Fourier transform, we present several classes of examples of this phenomenon.