Locally compact quantum groupoid

TL;DR

This paper develops a duality theory for locally compact quantum groupoids, connecting them with measured quantum groupoids, and provides examples illustrating these structures.

Contribution

It introduces a duality framework for locally compact quantum groupoids and establishes their relationship with measured quantum groupoids, expanding the theoretical understanding.

Findings

Association of locally compact quantum groupoids with measured quantum groupoids

Construction of a duality for locally compact quantum groupoids

Examples demonstrating the theory

Abstract

The theory of measured quantum groupoids, as defined by Lesieur and myself, was made to generalize the theory of quantum groups made by Kustarmans and Vaes, but was only defined in a von Neumann algebra setting; Th. Timmermann constructed locally compact quantum groupoids, which is a C*-version of quantum groupoids. Here, we associate to such a locally compact quantum groupoid a measured quantum groupoid in which it is weakly dense; we then associate to a measured quantum groupoid a locally compact quantum groupoid which is weakly dense in the measured quantum groupoid, but such a locally compact quantum groupoid may be not unique; we construct a duality of locally compact quantum groupoids. We give then examples of locally compact quantum groupoids.