A remark on twists and the notion of torsion-free discrete quantum groups

TL;DR

This paper investigates twists of reduced locally compact quantum groups, establishing duality properties and torsion-freeness, especially for discrete, torsion-free quantum groups and cocycle twists of Lie groups.

Contribution

It introduces twisted dual coactions satisfying Takesaki-Takai duality and shows that such twists preserve torsion-freeness in quantum groups.

Findings

Twisted dual coactions satisfy a form of Takesaki-Takai duality.

Twists of discrete, torsion-free quantum groups remain torsion-free.

Cocycle twists of simply connected Lie groups are torsion-free.

Abstract

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki-Takai duality. The twisted Takesaki-Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld-Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.