\'Equivalence mono\"idale de groupes quantiques et K-th\'eorie bivariante

TL;DR

This paper extends the concept of monoidal equivalence to regular locally compact quantum groups, establishing categorical equivalences for their actions on C*-algebras and K-theory, using duality theorems and quantum groupoids.

Contribution

It introduces an induction procedure and categorical equivalences for actions of monoidally equivalent quantum groups on C*-algebras and K-theory.

Findings

Established an equivalence of categories ${A}^{G_1}$ and ${A}^{G_2}$ for quantum group actions.

Derived a canonical equivalence of categories ${KK}^{G_1}$ and ${KK}^{G_2}$.

Extended duality theorems to actions of measured quantum groupoids.

Abstract

In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_1$ and $G_2$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories ${A}^{G_1}$ and ${A}^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on $C^*$-algebras. As an application of this result, we derive a canonical equivalence of the categories ${KK}^{G_1}$ and ${KK}^{G_2}$. We introduce and investigate a notion of actions on $C^*$-algebras of measured quantum groupoids on a finite basis. The proof of the equivalence between ${KK}^{G_1}$ and ${KK}^{G_2}$ relies on a version of the Takesaki-Takai duality theorem for continuous actions on $C^*$-algebras of measured quantum groupoids on a finite basis.