Twisting and Rieffel's deformation of locally compact quantum groups. Deformation of the Haar measure

TL;DR

This paper introduces a twisting construction for locally compact quantum groups that results in a non-trivial deformation of the Haar measure, providing new examples of quantum groups through Rieffel's deformation and duality.

Contribution

It extends previous work by incorporating Haar measure deformation and constructs new quantum groups via Rieffel's deformation and duality.

Findings

Deformation of Haar measure in quantum groups

Construction of new quantum group examples

Duality between twisting and Rieffel's deformation

Abstract

We develop the twisting construction for locally compact quantum groups. A new feature, in contrast to the previous work of M. Enock and the second author, is a non-trivial deformation of the Haar measure. Then we construct Rieffel's deformation of locally compact quantum groups and show that it is dual to the twisting. This allows to give new interesting concrete examples of locally compact quantum groups, in particular, deformations of the classical $az+b$ group and of the Woronowicz' quantum $az+b$ group.