The K-theory of free quantum groups

In this paper we study the K -theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K -amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K -theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ -element and that γ=1 . As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting

Quantum Field Theory on the Noncommutative Plane with Eq(2)E_q(2) Symmetry

We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with $E_q(2)$ quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the $E_q(2)$-invariant measure on the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally, we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the $E_q(2)$ quantum group.Comment: LaTeX, 26 page

Equivariant comparison of quantum homogeneous spaces

We prove the deformation invariance of the quantum homogeneous spaces of the q-deformation of simply connected simple compact Lie groups over the Poisson-Lie quantum subgroups, in the equivariant KK-theory with respect to the translation action by maximal tori. This extends a result of Neshveyev-Tuset to the equivariant setting. As applications, we prove the ring isomorphism of the K-group of Gq with respect to the coproduct of C(Gq), and an analogue of the Borsuk-Ulam theorem for quantum spheres.Comment: 21 page

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres

Extensions and degenerations of spectral triples

For a unital C*-algebra A, which is equipped with a spectral triple and an extension T of A by the compacts, we construct a family of spectral triples associated to T and depending on the two positive parameters (s,t). Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact operators is obtained under the limit - in this metric - for (s,1) -> (0, 1), while the basic spectral triple, associated to A, is obtained from this family under a sort of a dual limiting process for (1, t) -> (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes' 7 axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s sphere plus comments on the relations to matricial quantum metrics. In ver.3 the word "deformations" in the original title has changed to "degenerations" and some illustrative remarks on this aspect are adde

A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries

Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde

Classification of minimal actions of a compact Kac algebra with amenable dual

We show the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II$_1$. This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument.Comment: 68 pages, Introduction rewritten; minor correction

Projective Fourier Duality and Weyl Quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures. requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text improvements and crucial typos correction

New methodology for optimizing transit priority at the network level

A new methodology for optimizing transit road space priority at the network level is proposed. Transit vehicles carry large numbers of passengers within congested road space efficiently. This aids justification of transit priority. Almost all studies that have investigated transit priority lanes focus at a link or an arterial road level, and no study has investigated road space allocation for priority from a network perspective. The aim of the proposed approach is to find the optimum combination of exclusive lanes in an existing operational transport network. Mode share is assumed variable, and an assignment is performed for both private and transit traffic. The problem is formulated by using bilevel programming, which minimizes the total travel time. The approach is applied to an example network and the results are discussed. The approach can identify the optimal combination of transit priority lanes and achieve the global optimum of the objective function. Areas for further development are discussed