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.
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 quantum groups used as building ingredients of the bicrossed product. Finally, we give examples of non-amenable locally compact quantum groups obtained by a bicrossed product construction.
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 Woronowicz are much more difficult to construct. We will make use of the bicrossed product construction to obtain our examples. This construction dates back to Kac, Takeuchi and Majid. It was also considered by Baaj & Skandalis.
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 is to analyze and collect enough information from a Poisson-Lie group, and using it to carry out a ``cocycle bicrossed product construction''. Constructions are done using multiplicative unitary operators, obtaining C*-algebraic, locally compact quantum (semi-)groups.
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 to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.
We establish two conditions equivalent to coamenability for type I locally compact quantum groups. The first condition is concerned with the spectra of certain convolution operators on the space $\operatorname{L}^2(\operatorname{Irr}(\mathbb{G}))$ of functions which are square integrable with respect to the Plancherel measure. The second condition involves spectra of character-like operators associated with direct integrals of irreducible representations. As examples we study special classes of quantum groups: classical, dual to classical, compact or given by a certain bicrossed product constr…
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 these double crossed products and several links between double crossed products and bicrossed products. In an appendix, we study the Radon-Nikodym derivative of a weight under a quantum group action, following Yamanouchi and obtain, as a corollary, a new characterization of closed quantum subgroups.
Beginning with a skew-symmetric matrix, we define a certain Poisson--Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or "Lie-Poisson") Poisson bracket. By analyzing this Poisson structure, we gather enough information to construct a C*-algebraic locally compact quantum group, via the "cocycle bicrossed product construction" method. The quantum group thus obtained is shown to be a deformation quantization of the Poisson-Lie group, in the sense of Rieffel.
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…
In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\wid…
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 semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C*-algebras. On the way, we provide several remarks on C*-algebraic properties of quantum groups and their actions.
We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are …
We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (facto…
The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions. We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.
Quantization of locally compact groups associated with essentially bijective $1$-cocycles
Université Catholique de Louvain（天主教鲁汶大学）；Université de Reims Champagne-Ardenne（兰斯香槟-阿登大学）；University of Oslo（奥斯陆大学）；OsloMet - storbyuniversitetet（奥斯陆城市大学）
We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coincidi…
We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan's Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and Sołtan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are: (i) All finite-dimensiona…
We show that Haefliger's cohomology for etale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.
It shown that an a locally injective surjection on a compact metric space admits a canonical locally homeomorphic extension such that the associated C*-algebras are isomorphic. This is then used in a study of the possible inverse temperatures of KMS states for a generalized gauge action.
This article extends the main results of the publication arXiv:2001.01312 to the case of a twisted groupoid. More precisely, it gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal subgroupoid. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra. This is applied to C*-algebraic deformation quantization and to multiplier representations of locally compact abelian groups.
We consider the following problem. Suppose $α$ is an action of a locally compact group $G$ on a $C^*$-algebra $A$, $H$ is a closed subgroup of $G$, and $(π,U)$ is a covariant representation of $(A,H,α)$. For which closed subgroups $K$ containing $H$ is there a covariant representation $(π,V)$ of $(A,K,α)$ such that $V|_H=U$? We answer this problem by providing a criterion involving the induced representation of $(A,G,α)$. We then consider the dual problem for coactions of locally compact groups.
We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and anal…
In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the following extension of a result due to N. Brown and E. Guentner: Every locally compact second countable $G$ admits a proper affine action on the reflexive and strictly convex Banach space $\bigoplus^{\infty}_{n=1} L^{2n}(G, dμ),$ where the direct sum is taken in the $l^2$-sense.
The purpose of this paper is extend the notion of morphism of groupoids introduced by Zakrzewski to locally compact $σ$-compact groupoids endowed with Haar systems and to use the extension to construct a covariant functor from this category to a category of $C^{\ast}$-algebras in the sense of Woronowicz.
Using methods coming from non-formal equivariant quantization, we construct in this short note a unitary dual 2-cocycle on a discrete family of quotient groups of subgroups of the affine group of a local field (which is not of characteristic 2, nor an extension of $\mathbb{Q}_2$). Using results of De Commer about Galois objects in operator algebras, we obtain new examples of locally compact quantum groups in the setting of von Neumann algebras.
