Related papers: Quantum bohr compactification
We introduce the notion of self-similarity for compact quantum groups. For a finite set $X$, we introduce a $C^*$-algebra $\mathbb{A}_X$, which is the quantum automorphism group of the infinite homogeneous rooted tree $X^*$. Self-similar…
In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a…
We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…
We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the…
For a completely Hausdorff quasi-topological group $G$, we construct a universal pro-$C^*$-algebra $C(E^+G)$ as the non-commutative geometer's analogue of the total space $EG$ of the classifying principal $G$-bundle $EG\to BG$. The…
In recent papers and books, a global quantization has been developed for unimodular groups of type I. It involves operator-valued symbols defined on the product between the group $\mathsf{G}$ and its unitary dual $\widehat{\mathsf{G}}$,…
We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We…
In this paper we study of the BGG-categories $\mathcal O_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal O$ for a semisimple complex Lie algebra carry over to the quantum case. Of…
We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors…
A detailed account of the construction of a homogeneous space for the quantum "az+b" group is presented. The homogeneous space is described by a commutative C*-algebra which means that it is a classical space. Then a covariant differential…
The q-monopole bundle introduced previously is extended to a general construction for quantum group bundles with non-universal differential calculi. We show that the theory applies to several other classes of bundles as well, including…
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…
To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors…
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…
The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar…
We introduce two new formulations for the notion of "quantum metric on noncommutative space". For a compact noncommutative space associated to a unital C*-algebra, our quantum metrics are elements of the spatial tensor product of the…
We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $\mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible…
We construct inclusions of the form $(B_0\otimes P)^G\subset (B_1\otimes P)^G$, where $G$ is a compact quantum group of Kac type acting on an inclusion of finite dimensional $\c^*$-algebras $B_0\subset B_1$ and on a $II_1$ factor $P$. Under…
Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a…
This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the `Planck scale quantum group' $C[x]\bicross C[p]$ and its observable-state…