The Poisson-Fourier Transform for bicrossed products I: Abelian approximations and the quantum duality principle
A. Massar
TL;DR
The paper develops a framework to realize Drinfel'd's quantum duality principle for bicrossed products of locally compact groups by introducing abelian approximations and the Poisson--Fourier transform. It constructs a unitary PF transform that links quantizations of matched extensions G and H, enabling an isomorphism between the dual quantum groups of their quantizations. The approach combines cocycle bicrossed products, partial Fourier transforms, and careful matching of abelian approximations to produce explicit quantizations, illustrated through ax+b, Heisenberg, nilpotent, reductive homogeneous spaces, and a 3D example. This work broadens the operator-algebraic realization of quantum duality and provides a practical toolkit to generate and compare new quantum group quantizations arising from bicrossed structures.
Abstract
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 $η_G : G \to \hat W$ and $η_H : H \to \hat V$ 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.