A locally compact quantum group arising from quantization of the affine group of a local field

TL;DR

This paper constructs new examples of locally compact quantum groups from the affine group of a local field using equivariant quantization and operator algebra techniques, expanding the known classes of quantum groups.

Contribution

It introduces a novel method to produce locally compact quantum groups from quotient groups of affine groups over local fields, not of characteristic 2 or extensions of $ ext{ extbf{Q}}_2$, via dual 2-cocycles.

Findings

Constructed a unitary dual 2-cocycle on quotient groups of affine groups.

Produced new examples of locally compact quantum groups in von Neumann algebra setting.

Applied equivariant quantization and Galois object results to quantum group construction.

Abstract

Using methods coming from non-formal equivariant quantization, we construct in this short note a unitary dual 2-cocycle on a discrete family of quotient groups of subgroups of the affine group of a local field (which is not of characteristic 2, nor an extension of $\mathbb{Q}_2$). Using results of De Commer about Galois objects in operator algebras, we obtain new examples of locally compact quantum groups in the setting of von Neumann algebras.