Extensions of locally compact quantum groups and the bicrossed product construction

TL;DR

This paper develops the cocycle bicrossed product construction for locally compact quantum groups, establishing a correspondence with cleft extensions and providing new examples of such quantum groups.

Contribution

It introduces a new cocycle bicrossed product construction for locally compact quantum groups and links it to cleft extensions, expanding the class of known quantum groups.

Findings

Established a one-to-one correspondence between cocycle bicrossed products and cleft extensions

Developed the cocycle bicrossed product construction from matched pairs of quantum groups

Produced new examples of locally compact quantum groups

Abstract

In the framework of locally compact quantum groups, we study cocycle actions. We develop the cocycle bicrossed product construction, starting from a matched pair of locally compact quantum groups. We define exact sequences and establish a one-to-one correspondence between cocycle bicrossed products and cleft extensions. In this way, we obtain new examples of locally compact quantum groups.