The bicrossed product construction for locally compact quantum groups

The cocycle bicrossed product construction allows certain freedom in producing examples of locally compact quantum groups. We give an overview of some recent examples of this kind having remarkable properties

On Low-Dimensional Locally Compact Quantum Groups

Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact quantum groups by the cocycle bicrossed product construction. We also clarify the notion of an extension of locally compact quantum groups by relating it to the concept of a closed normal quantum subgroup and the quotient construction. Finally, we describe the infinitesimal objects of locally compact quantum quantum groups with 2 and 3 generators - Hopf *-algebras and Lie bialgebras.Comment: 64 pages, LaTeX, needs class-file irmadegm.cls. To appear in Locally Compact Quantum Groups and Groupoids. Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21 - 23, 200

A locally compact quantum group of triangular matrices

We construct a one parameter deformation of the group of $2\times 2$ upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual \cs-algebra and the dual comultiplication

On Z/2Z-extensions of pointed fusion categories

We give a classification of Z/2Z-graded fusion categories whose 0-component is a pointed fusion category. A number of concrete examples is considered.Comment: This article will be published by the Banach Center Publication

A locally compact quantum group of triangular matrices

We construct a one parameter deformation of the group of 2×2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual C∗-algebra and the dual comultiplication.Побудовано однопараметричну деформацію групи верхніх трикутних матриць розміру 2 × 2 із детермінантом 1 з використанням конструкції скруту. Цікавою рисою цього нового прикладу локально компактної квантової групи є нетривіальна деформація міри Хаара. Наведено також повний опис дуальної C*-алгебри та дуальної комультиплікації