# Quantization of subgroups of the affine group

## TL;DR

This paper explores the quantization of certain affine groups, establishing three equivalent methods, and reveals new cohomological phenomena in the structure of locally compact groups and their quantum counterparts.

## Contribution

It introduces three equivalent quantization approaches for affine groups with free orbits, linking them to Galois objects and cohomology classes, and extends finite group results to locally compact groups.

## Key findings

- Equivalence of three quantization methods for affine groups.
- Identification of a new cohomological phenomenon where the trivial class yields nontrivial Galois objects.
- Extension of Galois object parametrization to locally compact groups using recent cohomology results.

## Abstract

Consider a locally compact group $G=Q\ltimes V$ such that $V$ is abelian and the action of $Q$ on the dual abelian group $\hat V$ has a free orbit of full measure. We show that such a group $G$ can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of $G$ on $L^2(V)$; (2) by twisting the coproduct on the group von Neumann algebra of $G$ by a dual $2$-cocycle obtained from the $G$-equivariant Kohn-Nirenberg quantization of $V\times\hat V$; (3) by considering the bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$ both isomorphic to $Q$. In the simplest case of the $ax+b$ group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum $ax+b$ group of Baaj-Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups $G$. Using recent results of De Commer we show that a class of $G$-Galois objects is parametrized by certain cohomology classes in $H^2(G;\mathbb T)$. This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in $H^2(G;\mathbb T)$ can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group $G$ with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual $2$-cocycles such that the corresponding quantization of $G$ is neither commutative nor cocommutative.