1999/2001 2002/2003 2003/2004 2004/2005 2005/2006 2006/2007 2007/2008 2008/2009 2009/2010 2010/2011

10 October 2011

Given a Poisson action of a Poisson-Lie group on a Poisson manifold, there is a well understood notion of the moment map and the associated Poisson reduction generalizing the classical Hamiltonian reduction of Marsden and Weinstein. The subject of this talk is the question of generalizing this to the case of quantum groups. We will describe what a good candidate for the "quantum moment map" is and give a few examples illustrating what happens with the reduction in this setting.

17 October 2011

Motivated by classical circle bundles, we study spectral triples over the total space of noncommutative principal

3 November 2011 (Banach Center research group. Joint Noncommutative Geometry and Operator Algebras and Quantum Groups seminar. Exceptional time: Thursday 13:15.)

I will discuss a product system over the multiplicative semigroup of positive integers of Hilbert bimodules and show how the associated Nica-Toeplitz algebra is related to the C*-algebra

7 November 2011 (Banach Center research group.)

Let

7 November 2011 (Banach Center research group. Exceptional time: 14:15.)

We prove that quantum homogeneous spaces given by Poisson-Lie quantum subgroups of the

14 November 2011

It is well known that a locally compact group can be recovered from the category of its unitary representations equipped with the natural operations of sum and tensor product. I will explain that a similar approach fails when restricted to finite-dimensional representations, unless the group is virtually abelian. In particular, even for a non-abelian free group (which is residually finite), the natural bi-dual group build on finite-dimensional representations is strictly larger than the free group. This solves an old problem due to Hsin Chu and the methods answer several questions about the Bohr topology of the free group.

14 November 2011 (Exceptional time: 14:15.)

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra

21 November 2011

The quantum permutation group was introduced by S. Wang as the universal compact quantum group acting on a finite set

19 December 2011

A Hopf image of a given representation of a Hopf algebra

19 December 2011 (Exceptional time: 14:15.)

A general criterion for the stable freeness of the finitely generated projective modules associated to principal comodule algebras via one-dimensional corepresentations will be proved. The line bundles over quantum real and complex projective spaces of an arbitrary dimension

9 January 2012

The notion of a closed quantum subgroup of a given locally compact quantum group was introduced by S. Vaes. His definition uses a mixture of the C*-reduced, C*-universal, and von Neumann algebraic version of a given quantum group and its dual. Recently, S.L. Woronowicz proposed a definition based on the notion of a bicharacter and the concept of the C*-algebra generated by a quantum family of elements. In my talk, I shall give a number of equivalent characterizations of Woronowicz's definition and make a link between the Vaes and Woronowicz approaches. (Joint work with Matthew Daws, Adam Skalski and Piotr M. Sołtan.)

9 January 2012 (Exceptional time: 14:15.)

We study the correspondence between the unitary representations of transformation groupoids and systems of imprimitivity. Next, for the general case of locally compact transitive groupoids, we define representations induced by a representation of the isotropy subgroupoid and prove an imprimitivity theorem. These results generalize classical concepts of Mackey known in the representation theory of locally compact groups.

16 January 2012 (Banach Center research group.)

This talk concerns the K-theory of free quantum groups in the sense of Wang and Van Daele. More precisely, we show that the free products of free unitary and free orthogonal quantum groups are K-amenable, and establish an analogue of the Pimsner-Voiculescu exact sequence. As a particular consequence, we obtain an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of Baum-Connes conjecture methods to the framework of discrete quantum groups. It is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result, we show that the gamma-element of any free quantum groups equals 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. (Joint work with Roland Vergnioux.)

16 January 2012 (Banach Center research group. Exceptional time: 14:15.)

The Bowers and Stephenson conformally regular pentagonal tiling of the plane enjoys remarkable combinatorial and geometric properties. Since it does not have finite local complexity in any usual sense, it is beyond the standard tiling theory. On the other hand, the tiling can be completely described by its combinatorial data that, rather automatically, has finite local complexity. With the aim to compute its K-theory, we construct the hull and C*-algebra of this tiling solely from its combinatorial data. As the tiling possesses no natural

27 February 2012

Let

5 March 2012 (Banach Center research group.)

By a theorem due to Hoegh-Krohn, Landstad and Stormer, the isotypical components of an ergodic action of a compact group on a unital C*-algebra are finite dimensional. This was later generalized to compact quantum groups by F. Boca. On the other hand, since recently we know that the isotypical components of a free action of a compact quantum group on a unital C*-algebra are finitely-generated modules over the fixed-point subalgebra. In this talk, we give several characterizations, in terms of Galois maps, of actions for which such a finite generation property holds in the general setting of compact quantum groups acting on unital C*-algebras. In the classical case, they are precisely the actions whose field of isotropy groups is continuous. This is joint work with M. Yamashita, and a joint project with P.F. Baum and P.M. Hajac.

5 March 2012 (Banach Center research group. Exceptional time: 14:15.)

There are two special classes of compact quantum groups, namely the classical compact groups and the Fourier duals of discrete groups. In this talk, we discuss what can be said about the discrete group duals sitting inside a given compact quantum group. We will consider several examples such as Wang's quantum automorphism groups and Goswami's quantum isometry groups of connected compact Riemannian manifolds. This is joint work with T. Banica and J. Bhowmick.

12 March 2012

Spectral triples are a refinement of K-homology cycles modelled on the Dirac operator of a compact spin manifold. There are various regularity properties of spectral triples connected to dimension and Rieffel's notion of a noncommutative or quantum metric space. We are mostly interested in existence results for spectral triples with good properties, e.g. defining good metrics on state spaces, which may be regarded as noncommutative metrisation results. We will review some old and new results and constructions of spectral triples especially on crossed products. This is joint work with Andrew Hawkins, Adam Skalski and Stuart White.

12 March 2012 (Exceptional time: 14:15.)

Although the main part of this talk will have purely combinatorial/free-probabilistic nature, links to quantum groups will be described towards the end of the presentation. We will discuss certain combinatorial problems related to particular categories of partitions. Non-crossing partitions are fundamental combinatorial tools of free probability. In this talk, we will discuss their extended, so-called "coloured", versions. These naturally arise in relation to intertwiners of some unitary matrices. They in turn are related to the representation theory of some compact quantum symmetry groups. (Most presented results come from joint work with Teo Banica.)

19 March 2012

As discovered by Shuzhou Wang, the universal action of a compact quantum group on the set of

26 March 2012

Kadison and Kastler equipped the set of all C*-subalgebras of

26 March 2012 (Exceptional time: 14:15.)

Since the 1950ies there has been a close connection between special functions and Lie groups, and this cross fertilization has turned out to be very fruitful for both sides. Even today this relation is a source of new and interesting results. Several aspects of this interplay found their generalization to appropriate relations between quantum groups and special functions of basic hypergeometric type almost immediately after the introduction of quantum groups at the end of the 1980ies. Already compact quantum groups led to new and interesting results for special functions, including, e.g., addition and product formulae which would have been hard to obtain otherwise. Later, special functions were used in the construction of certain non-compact quantum groups. In particular, this concerns the quantum analogue of the normalizer of

2 April 2012

Bieberbach manifolds are compact quotients of

23 April 2012

Let

23 April 2012 (Exceptional time: 14:15.)

Let

7 May 2012 (Banach Center research group.)

For well over a decade it has been an open problem to find a

7 May 2012 (Banach Center research group. Exceptional time: 14:15.)

The class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. This allows us to go beyond the category of comodule algebras when constructing examples of principal coactions. The aim of this talk is to show such an example by constructing a family of coalgebraic noncommutative deformations of the

28 May 2012

For a Lie group

28 May 2012 (Exceptional time: 14:15.)

We will discuss how KK-theory can be constructed via correspondences of spectral triples. This involves the notion of a connection analogous to a vector bundle connection. By factoring a spectral triple into a correspondence applied to a spectral triple over a commutative base, we can describe the gauge theory of the initial spectral triple in terms of connections. We will exemplify this construction on noncommutive tori and quantum Hopf fibrations. (Joint work with S. Brain.)