NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Śniadeckich 8, room 322, Mondays, 10:15-12:00



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



10 October 2011

TOWARDS A QUANTUM MOMENT MAP

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.

RYSZARD NEST (Københavns Universitet, Denmark)



17 October 2011

NEW DIRAC OPERATORS THROUGH NONCOMMUTATIVE CIRCLE BUNDLES

Motivated by classical circle bundles, we study spectral triples over the total space of noncommutative principal U(1)-bundles. We propose a compatibility condition between a connection and a Dirac operator, and analyze it in detail on noncommutative three-tori. Thus we find a family of new Dirac operators that arise from the base-space Dirac operator and a suitable connection. For noncommutative two-tori, we obtain a restricted version of the Gauss-Bonnet theorem. This is joint work with Ludwik Dabrowski.

ANDRZEJ SITARZ (IMPAN/Uniwersytet Jagielloński)



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

THE CUNTZ ALGEBRA Q_N AND C*-ALGEBRAS OF PRODUCT SYSTEMS

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 Q_N introduced recently by Cuntz.

WOJCIECH SZYMAŃSKI (Syddansk Universitet, Odense, Denmark)



7 November 2011 (Banach Center research group.)

PRINCIPAL COMPACT QUANTUM GROUP ACTIONS ON UNITAL C*-ALGEBRAS ARE GALOIS

Let G be a compact group acting on a compact Hausdorff space. Ellwood observed that one can easily characterise the principality of this action on the level of the associated C*-algebras by means of a density condition. More recently, Baum and Hajac showed that Ellwood's condition is equivalent to the principality of the associated Peter-Weyl comodule algebra, and thus embedded the category of compact Hausdorff principal bundles into the category of comodule algebras with Galois coactions. It is the aim of this talk to prove that the above notions are still equivalent when G is a compact quantum group acting on a unital C*-algebra. (An important special case G=U(1) was proven much earlier by W.Szymański.) The key new element in our approach, inspired by the work of S.Popa and A.Wassermann on ergodic compact group actions on von Neumann algebras, is the use of Pimsner-Popa type inequalities to allow passages between different Hilbert C*-modules associated to an action of a compact quantum group. (Part of a joint project with P.F.Baum, P.M.Hajac and W.Szymański.)

KENNY DE COMMER (Université de Cergy-Pontoise, France)



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

EQUIVARIANT COMPARISON OF QUANTUM HOMOGENEOUS SPACES

We prove that quantum homogeneous spaces given by Poisson-Lie quantum subgroups of the q-deformations of simply connected simple compact Lie groups are equivariantly KK-equivalent to the classical ones. This result extends the nonequivariant case of Neshveyev and Tuset. As an application, we obtain an analogue of the Borsuk-Ulam theorem for quantum spheres conjectured by Baum and Hajac.

MAKOTO YAMASHITA (Università di Roma Tor Vergata, Italy)



14 November 2011

CONVERGENT SEQUENCES IN DISCRETE GROUPS AND CHU DUALITY

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.

ANDREAS THOM (Universität Leipzig, Germany)



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

ALGEBRAIC GEOMETRY OF TOPOLOGICAL SPACES

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem of Joseph Gubeladze, that is, we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parameterized version of the celebrated theorem, proved independently by Daniel Quillen and Andrei Suslin, that all finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case when M=Z^n.

ANDREAS THOM (Universität Leipzig, Germany)



21 November 2011

NONEXISTENCE OF NONCLASSICAL QUANTUM PERMUTATIONS ACTING ON MAPS

The quantum permutation group was introduced by S. Wang as the universal compact quantum group acting on a finite set X in a way preserving the counting measure of X. In the classical case, the counting measure can be interpreted as a function on the set P(X) of all subsets of X that is invariant under the action of the permutation group on P(X) induced from its action on X. More generally, the permutations of X naturally act on Map(X,Y) for any set Y. In particular, for Y={0,1} one obtains the action of permutations on P(X). However, we will show that quantum permutations of X inducing a quantum action on spaces of maps Map(X,Y) must be necessarily classical. On the other hand, allowing "quantum maps" acted on by quantum permutations one faces the fact that classically X = Map({0},X) is itself a set of maps. Therefore, X itself should also be quantized in an appropriate sense going beyond the context of Wang's construction. This would generalize the concept of a quantum family of maps studied by P.M. Sołtan.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)



19 December 2011

HOPF IMAGES, IDEMPOTENT STATES AND MATRIX MODELS OF QUANTUM GROUPS

A Hopf image of a given representation of a Hopf algebra A is the largest Hopf quotient of A through which the representation factorises in a natural way. This notion was introduced by T. Banica and J. Bichon in 2010. If the Hopf image is equal to A, the considered representation is called inner faithful. The question of the existence of an inner faithful finite-dimensional representation of a given Hopf algebra is a natural counterpart of the investigation of linearity of a given discrete group G. Hence Hopf algebras admitting such representations are called inner linear. In this talk, we will recall the theory developed by Banica and Bichon, present a new approach to Hopf images of compact quantum groups via the theory of idempotent states developed mainly by U. Franz and A. Skalski, formulate some open problems related to inner linearity, and discuss connections to various notions of matrix models of quantum groups. (Joint work with Teodor Banica and Uwe Franz.)

ADAM SKALSKI (IMPAN)



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

A CRITERION FOR STABLE TRIVIALITY OF THE ASSOCIATED NONCOMMUTATIVE LINE BUNDLES

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 n will serve as examples. Here the criterion allows one to conclude that the stable non-freeness in the general case follows from the stable non-freeness for n=2 and n=1, respectively. The real case for n=2 was handled by direct K-theory methods, and the complex case for n=1 by the noncommutative index pairing. In both cases the line bundles associated by the non-trivial (non-zero winding number) corepresenations were shown to be not stably free. Hence all such non-zero winding number line bundles over these quantum projective spaces are not stably free.

PIOTR M. HAJAC (IMPAN/Uniwersytet Warszawski)



9 January 2012

CLOSED QUANTUM SUBGROUPS

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.)

PAWEŁ Ł. KASPRZAK (Uniwersytet Warszawski/IMPAN)



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

REPRESENTATIONS OF GROUPOIDS AND IMPRIMITIVITY SYSTEMS .

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.

LESZEK PYSIAK (Politechnika Warszawska)



16 January 2012 (Banach Center research group.)

THE K-THEORY OF FREE QUANTUM GROUPS

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.)

CHRISTIAN VOIGT (Universität Münster, Germany)



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

INVARIANTS FOR A CONFORMALLY REGULAR PENTAGONAL TILING OF THE PLANE

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 R^2 action by translation, there is no a priori reason to expect that the K-theory of the C*-algebra of the tiling is the same as the K-theory or cohomology of the hull of the tiling, and it would be very interesting if they were different.

MARIA RAMIREZ-SOLANO (Københavns Universitet, Denmark)



27 February 2012

EXACT CROSSED-PRODUCTS AND A COUNTER-EXAMPLE (TO BAUM-CONNES WITH COEFFICIENTS) REVISITED

Let G be a locally compact Hausdorff topological group which is second countable. Let G-C* denote the category of all G-C* algebras. The morphisms in G-C* are *-homomorphisms which are G-equivariant. Consider crossed-products which are intermediate between the max crossed-product and the reduced crossed-product. Such a crossed-product is said to be exact if and only if, whenever 0 -> A -> B -> C -> 0 is a short exact sequence in G-C*, the sequence of C* algebras obtained by applying the crossed-product is exact. For example, the max crossed product is exact for any G. The reduced crossed-product is exact if and only if G is exact. P. Baum conjectured and E. Kirchberg proved that there always exists a unique minimal exact crossed-product. The Baum-Connes (BC) conjecture with coefficients should then be reformulated to state that the BC map K_*(E_G,A) -> K_*(C*(G,A)) is an isomorphism, where C*(G,A) is the minimal exact crossed-product. In this reformulated version of BC with coefficients it is probable that the Higson-Lafforgue-Skandalis counter-example is no longer a counter-example.

PAUL F. BAUM (Pennsylvania State University, State College, USA / IMPAN)



5 March 2012 (Banach Center research group.)

A FINITENESS CONDITION FOR COMPACT QUANTUM GROUP ACTIONS

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.

KENNY DE COMMER (Université de Cergy-Pontoise, France)



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

GROUP-DUAL SUBGROUPS OF COMPACT QUANTUM GROUPS

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.

KENNY DE COMMER (Université de Cergy-Pontoise, France)



12 March 2012

CONSTRUCTING SPECTRAL TRIPLES ON C*-ALGEBRAS

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.

JOACHIM ZACHARIAS (The University of Nottingham, England)



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

CATEGORIES OF COLOURED PARTITIONS RELATED TO REPRESENTATIONS OF COMPACT QUANTUM GROUPS

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.)

ADAM SKALSKI (IMPAN)



19 March 2012

QUANTUM PERMUTATIONS OF TWO ELEMENTS

As discovered by Shuzhou Wang, the universal action of a compact quantum group on the set of n elements non-trivially extends the action of the classical permutation group beginning at n=4. This fact has been generalized from compact quantum groups to cosemisimple or involutive Hopf algebras by Julien Bichon. Now we show that for the wider class of Hopf algebras with bijective antipode non-classical quantum permutations start to appear already at n=2. It is also the case for the class of all Hopf algebras because the universal property then has a solution in terms of Manin's Hopf envelope. However, unlike Manin's construction, the Hopf algebra admitting a coaction (on the algebra of functions on the set of n elements) that is universal in the class of Hopf algebras with bijective antipode is presented as a localization of a finitely generated algebra.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)



26 March 2012

KADISON-KASTLER STABILITY FOR OPERATOR ALGEBRAS

Kadison and Kastler equipped the set of all C*-subalgebras of B(H) with a natural metric and conjectured that sufficiently close algebras should be isomorphic. This is a uniform metric: two algebras A and B are close in the Kadison-Kastler metric if every operator in the unit ball of A can be well approximated in the unit ball of B and vice versa. Kadison and Kastler's conjecture was established in the 70's when one algebra is an injective von Neumann algebra. In this talk, I will discuss some recent progress for non-injective von Neumann algebras (mainly arising as crossed products) and the connections between this problem and the similarity problem.

STUART WHITE (The University of Glasgow, Scotland)



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

QUANTUM GROUPS AND SPECIAL FUNCTIONS

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 SU(1,1) in SL(2,C). The harmonic analysis on this quantum group has given rise to some new notions and results for the associated special functions. In this talk, the general development of the relation between compact quantum groups and special functions will be shortly reviewed. Then we extend the picture to include non-compact quantum groups. In particular, we will consider the aforementioned example of a quantum-group normalizer and its relation to special functions. We will end by discussing the status of a personal wish-list of results we desire for the interpretation of special functions on quantum groups.

ERIK KOELINK (Radboud Universiteit Nijmegen, the Netherlands)



2 April 2012

THE K-THEORY OF NONCOMMUTATIVE BIEBERBACH SPACES

Bieberbach manifolds are compact quotients of R^n by a free, properly discontinous and isometric action of a discrete group. The noncommutative counterparts of Bieberbach spaces in dimension 3 arise as quotients of a three-dimensional noncommutative torus by an action of a finite group. We compute the K-theory of all such noncommutative Bieberbach spaces. Our methods use the results (of Walters and Echterhoff et al) on unbounded twisted traces and on an explicit presentation of the K-theory generators of the quotients of a two-dimensional noncommutative torus by an action of a cyclic group.

PIOTR OLCZYKOWSKI (Uniwersytet Jagielloński)



23 April 2012

THE DUALITY OF GENERALIZED HOPF AND LIE ALGEBRAS

Let H be a Hopf algebra and P be the functor that assigns to each Hopf algebra the Lie algebra of its primitive elements. Michaelis introduced the notion of a Lie coalgebra and defined a dual functor Q that assigns to each Hopf algebra a Lie coalgebra. Moreover, he proved that P of the Sweedler dual of H is isomorphic with the space of functionals on Q(H). The aim of this talk is to show a more abstract version of the Michaelis theorem, so that it can be applied to a broader class of Hopf algebra type objects and dualities. (Joint work with Isar Goyvaerts.)

JOOST VERCRUYSSE (Université Libre de Bruxelles, Belgium)



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

CLEFT-TYPE BICOMODULES AND GALOIS CO-OBJECTS

Let B be a k-bialgebra. The "fundamental theorem for Hopf modules" can be stated as the equivalence of the following three properties: (i) B admits an antipode (i.e. B is a Hopf algebra); (ii) the canonical map for the coproduct coaction of B on itself is bijective; (iii) the category of k-modules is equivalent to the category of (right, right) B-Hopf modules through the functor of tensoring with B on the right. The three properties have been generalized in various ways and settings not necessarily preserving the equivalence between the properties (e.g., in the setting of corings and entwining structures). For instance, a "cleft" bicomodule is defined by means of a cleaving map that mimics the behaviour of the antipode, a "Galois comodule" has a bijective canonical map, and a "Galois co-object" always induces an equivalence of categories between k-modules and a generalized category of Hopf-modules. Our aim is to establish the equivalence of similar three conditions in the setting of (dual) Galois comodules. In particular, this will allow us to show that Galois co-objects are equipped with an analogue of the antipode map.

JOOST VERCRUYSSE (Université Libre de Bruxelles, Belgium)



7 May 2012 (Banach Center research group.)

NONCOMMUTATIVE DIFFERENTIAL GEOMETRY OF A QUANTUM DEFORMATION OF THE 7-DIMENSIONAL HOPF FIBRATION

For well over a decade it has been an open problem to find a q-deformed analogue of the principal SU(2) Hopf fibration of the 7-dimensional sphere equipped with finite-dimensional noncommutative differential structures. In the literature one finds a range of examples of such quantum `quaternionic' Hopf fibrations, but only at the level of infinite dimensional `universal' differential structures. In this talk I will present an example of a q-deformed quaternionic Hopf fibration whose total space and base space are equipped with finite-dimensional differential structures. By investigating the quantum symmetries of the fibration, I will describe the geometry of the corresponding quantum twistor space (i.e. a noncommutative 3-dimensional complex projective space) and use it to study a system of anti-self-duality equations on the base space quantum 4-dimensional sphere. The system of equations admits a canonical `instanton' solution coming from a natural projection defining a noncommutative vector bundle over this quantum sphere. (Joint work with G. Landi.)

SIMON BRAIN (Université du Luxembourg)



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

A NON-COMODULE-ALGEBRA EXAMPLE OF THE PULLBACK OF PRINCIPAL COACTIONS

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 U(1)-principal bundle S7 → CP3. (Joint work with P. M. Hajac.)

ELMAR WAGNER (Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico)



28 May 2012

THE BAAJ-SKANDALIS "ax+b" QUANTUM GROUP AS AN EXAMPLE OF A C*-ALGEBRAIC DRINFELD TWIST

For a Lie group G and three closed subgroups A, B, C, such that the intersection of A with B and C with B is trivial, G=AB and the intersection of BC with CB is open and dense in G, we construct a Drinfeld twist that can be used to twist the comultiplication of a certain quantum group related to the decomposition G=AB. We exemplify this construction by the Baaj and Skandalis "ax+b" quantum group and, time permitting, by the kappa-Poincare quantum group.

PIOTR STACHURA (Warsaw University of Life Sciences)



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

NONCOMMUTATIVE CORRESPONDANCES AND APPLICATIONS TO GAUGE THEORY

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.)

BRAM MESLAND (The University of Manchester, England)