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

TL;DR

This paper introduces the Poisson-Fourier transform for bicrossed product quantum groups, providing a method to realize the quantum duality principle explicitly in cases with abelian approximations.

Contribution

It constructs an explicit unitary operator implementing the quantum duality principle for certain bicrossed product quantum groups with abelian approximations.

Findings

Constructed the Poisson-Fourier transform as a unitary operator.

Established isomorphism between dual quantum groups via the transform.

Presented examples illustrating the duality phenomenon.

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 $\eta_G : G \to \hat W$ 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.