Modular properties of matrix coefficients of corepresentations of a locally compact quantum group

TL;DR

This paper derives formulas for the modular operator and conjugation using matrix coefficients of corepresentations in locally compact quantum groups, enabling explicit descriptions of modular automorphisms and Duflo-Moore operators.

Contribution

It provides new explicit formulas linking modular structures to matrix coefficients in quantum groups, advancing understanding of their harmonic analysis.

Findings

Formulas for modular operator and conjugation in terms of matrix coefficients

Expression of modular automorphism group via matrix coefficients for unimodular quantum groups

Determination of Duflo-Moore operators for a specific quantum group analogue

Abstract

We give a formula for the modular operator and modular conjugation in terms of matrix coefficients of corepresentations of a quantum group in the sense of Kustermans and Vaes. As a consequence, the modular autmorphism group of a unimodular quantum group can be expressed in terms of matrix coefficients. As an application, we determine the Duflo-Moore operators for the quantum group analogue of the normaliser of SU(1,1) in $SL(2,\mathbb{C}$).