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 ${\mathcal{G}}_{\hbar}$ of a Poisson-Lie group $\mathcal{G}$ should be dual as a quantum group to a quantization $\mathcal{G}^*_{\hbar}$ of the Poisson dual group $\mathcal{G}^*\!\!$. In this paper we consider pairs $(\mathcal{G} = G \ltimes V, \mathcal{G}^* = H \ltimes W)$ with $V, W$ abelian, where we can realise the quantizations ${\mathcal{G}}_{\hbar}$ and $\mathcal{G}^*_{\hbar}$ as a bicrossed product between $G$ and $H$ in the setting of locally compact quantum groups. Assuming the existence of suitable maps $\eta_G : G \to \hat W$ and $\eta_H : H \to \hat V$ which we call abelian approximations, we implement the quantum duality principle by constructing an explicit unitary operator $\mathcal{F}_{\mathcal{G}} : \mathrm{L}^2(\mathcal{G}) \to \mathrm{L}^2(\mathcal{G}^*)$, the Poisson-Fourier transform between $\mathcal{G}$ and $\mathcal{G}^*$. It induces an isomorphism of locally compact quantum group $\mathcal{F}_{\mathcal{G}} : \hat{\mathcal{G}}_{\hbar} \cong \mathcal{G}^*_{\hbar}$. After discussing the general framework for the Poisson-Fourier transform, we present several classes of examples of this phenomenon.