Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

On cohomology theory for topological groups

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A.     

Groupoid cohomology and extensions

We show that Haefliger's cohomology for étale 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.

     

Torsion in the cohomology of finite loop spaces and the Eilenberg–Moore spectral sequence

We use the Eilenberg–Moore spectral sequence to study torsion phenomena in the integral cohomology of finite loop spaces with maximal torus generalizing some results for compact Lie groups due to Kac.     

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.     

A Criterion for the Second Real Continuous Bounded Cohomology of a Locally Compact Group to be Finite-Dimensional

We present a brief review of the theory of quasi-characters and quasi-representations and prove a necessary and sufficient condition that the second real continuous bounded cohomology of a locally compact group to be finite-dimensional. This criterion is established by using the properties of continuous pseudocharacters on a locally compact group.     

The Weil-��tale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.     

Quiver varieties and Frenkel–Kac construction

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel–Kac construction from the latter.     

Structure Properties and Real Continuous Bounded 2-Cohomology of Locally Compact Groups

With the help of some structure results for locally compact groups, the second real continuous bounded cohomology group of a connected locally compact group is described and it is proved that the corresponding group is finite-dimensional for any almost connected locally compact group.     

Bredon-Illman cohomology with actions of totally disconnected groups

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

Equivariant Brauer and Picard groups and a Chase–Harrison–Rosenberg exact sequence

The classical Chase–Harrison–Rosenberg exact sequence relates the Picard and Brauer groups of a Galois extension S of a commutative ring R to the group cohomology of the Galois group. We associate to each action of a locally compact group G on a locally compact space X two groups which we call the equivariant Picard group and the equivariant Brauer group. We then prove an analogue of the Chase–Harrison–Rosenberg exact sequence in the which the roles of the Picard and Brauer groups are played by their equivariant analogues.     

Sur la structure des algèbres de Kac,I

We elaborate on the relative situation of a Kac algebra and its dual, then, introducing three invariants we obtain a generalization of Heisenberg's commutation relations. Using the operator-valued weights introduced by U. Haagerup, we give a simpler definition for Kac algebras. As an application, we generalize the unicity of the left Haar measure on a locally compact group.     

On (co)homology locally connected spaces

We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon.     

Embeddable quantum homogeneous spaces

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above.     

A cohomology approach to the extension problem for commutative hypergroups

The purpose of this paper is to determine all commutative hypergroup extensions of a countable discrete commutative hypergroup by a locally compact Abelian group, in terms of second order cohomology of hypergroups, a notion which generalizes the cohomology of groups.     

Equivariant sheaf cohomdlogy

In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)^G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology H_G(X;-).     

Cohomology of Hopf C*-Algebras and Hopf Von Neumann Algebras

We define two canonical cohomology theories for Hopf C^*-algebrasand for Hopf von Neumann algebras (with coefficients in theircomodules). We then study the situations when these cohomologiesvanish. The cases of locally compact groups and compact quantumgroups are considered in more detail. E-mail: c.k.ng{at}qub.ac.uk2000 Mathematical Subject Classification: primary 46L05, 46L55;secondary 43A07, 22D25.     

Localization of IC-complexes on Kashiwara’s flag scheme and representations of Kac–Moody algebras

We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara’s flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac–Moody algebras and give a new proof of the Kazhdan–Lusztig conjecture for blocks containing an antidominant element.