27 research outputs found
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
A locally compact quantum group of triangular matrices
We construct a one parameter deformation of the group of 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 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
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
Fusion modules and amenability of coideals of compact and discrete quantum groups
We give a definition of an amenable fusion module over a fusion algebra. A
notion of relative integrability for the `coduals' of coideals of compact
quantum groups was recently introduced in the joint work of de Commer and
Dzokou Talla. We use this property to construct an analogue of the
quasi-regular representation. Then, we characterize a certain coamenability
property of quasi-regular representations with amenability of their associated
fusion modules. Afterwards, we obtain a duality result that generalizes
Tomatsu's theorem for this coamenability property and an amenability property
of their `codual' coideals (under an additional assumption). As an example, we
apply this result to show the fusion modules associated to certain non-standard
Podle\'s spheres are amenable.Comment: 36 pages + references. v2: quasi-integrable term changed to
relatively integrable to change in cited work along with other minor changes
including the streamlining and elimination of some proof