2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60766In 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.66 pages, LaTeX. Some references added. Discussion added how examples are deformations of usual groupsOperator AlgebrasQuantum Algebratext