Related papers: The bicrossed product construction for locally com…
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…
We study stability properties of amenable locally compact quantum groups under the bicrossed product construction. We obtain as our main result an equivalence between amenability of the bicrossed product and amenability of the matched…
The main aim of this paper is to introduce some examples of non-compact locally compact quantum groups to a non-specialized audience. The major importance of these examples is their simplicity. Other examples as the quantum E(2) group of…
By working with several specific Poisson-Lie groups arising from Heisenberg Lie bialgebras and by carrying out their quantizations, a case is made for a useful but simple method of constructing locally compact quantum groups. The strategy…
For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld's quantum double construction. We study C*-algebraic properties of…
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…
We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a topological version of the Kac exact sequence and provide methods…
We consider bicrossed products obtained by twisting compact semi-direct products by a suitable finite subgroup. Under some restriction, we give a practical criterion for the discrete dual of such bicrossed products to have the rapid decay…
We investigate one question regarding bicrossed products of finite groups which we believe has the potential of being approachable for other classes of algebraic objects (algebras, Hopf algebras). The problem is to classify the groups that…
We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and and produce an easy…
The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some usefull properties of subgroups and amalgamated…
We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a…
In this article, we investigate the notion of a Galois object for a locally compact quantum group M. Such an object consists of a von Neumann algebra N equipped with an ergodic integrable coaction of M on N, such that the crossed product is…
In this paper, we study C*-algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not…
In this article, we give a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes).…
Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated…
We introduce the notion of a crossed product of an algebra by a coalgebra $C$, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra…
It is shown that quantum Euclidean groups $E_q(2)$, $E_\kappa(2)$ and $E_\kappa(3)$ have the structure of generalised crossed products.
Crossed products whose unit group is locally solvable
We introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition. We also establish a…