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


1 October 2010 (Exceptional time and place: Friday 14:30, room 105. Banach Center mini-school.)

NONCOMMUTATIVE KAPPA-MINKOWSKI SPACETIME

The main aim of this talk is to present the classification of unitary (regular) representations of the universal C*-algebra of the noncommutative kappa-Minkowski spacetime. We will also describe the Weyl symbolic calculus and relate it to the group algebra of the affine group ax+b. It will be shown that the uncertainty relations allow for states with all spacetime coordinates localised with arbitrary precision. Finally, we shall explain an extended model of the kappa-Minkowski spacetime which enjoys a unitary implementation of the Poincare group.

LUDWIK DĄBROWSKI (SISSA, Trieste, Italy)


11 October 2010

SPECTRAL TRIPLES FOR COMPACT RIEMANNIAN MANIFOLDS

Spectral triples for the noncommutative analogue of spin-c manifolds are standard lore, by now. Passage from spin-c to general (compact) Riemannian manifolds, and back where possible, requires a good notion of Morita equivalence for spectral triples. We introduce a noncommutative picture for Riemannian manifolds, and show that the relationship with noncommutative spin-c manifolds can be implemented precisely via products of unbounded Kasparov modules. (Joint work with Steven Lord and Adam Rennie.)

JOSEPH C. VÁRILLY (Universidad de Costa Rica)


11 October 2010 (Exceptional time: 14:15.)

ON A PROBLEM OF ATIYAH

In 1976, Michael Atiyah defined L2-Betti numbers for manifolds and asked a question about their rationality. These numbers arise as the von Neumann dimensions of kernels of certain operators acting on the L2-space of the fundamental group of a manifold. The problem concerning their values is closely related to the Kaplansky zero-divisor question. We present constructions of closed manifolds with irrational L2-Betti numbers.

ANDRZEJ ŻUK (Université Paris 7, France)


18 October 2010

HOW QUANTUM ISOMETRY GROUPS MET LIBERATED QUANTUM GROUPS AND PRODUCED NEW EXAMPLES

Last two-three years brought a fast development of the theory of quantum isometry groups of noncommutative manifolds, resulting in associating with any well-behaved spectral triple of compact type a compact quantum group in the sense of Stanisław L. Woronowicz. In particular, together with Jyotishman Bhowmick, we showed how the dual of each finitely generated discrete group admits a natural quantum isometry group and computed the resulting quantum group for the duals of free groups. As realised later, this example is closely connected to the liberation procedure (studied intensively by Teodor Banica and Roland Speicher) associating to classical orthogonal groups their free quantum counterparts. This connection led to the discovery of a new two-parameter family of quantum symmetry groups, which we will describe in detail in this talk. The main results to be presented are based on the very recent joint work with Teodor Banica.

ADAM SKALSKI (IMPAN)


25 October 2010

DUALITY FOR SIN-GROUPS

We suggest a generalization of the Pontryagin duality from Abelian locally compact groups to SIN-groups, i.e., groups with a base of invariant neighborhoods of identity. We aim to demonstrate the duality procedure that is based on the notion of a C*-envelope and does not require the Haar measure or its analogues. This procedure takes into account only norm continuous group representations, and this is why we are limited to the class of SIN-groups. The self-dual category into which we embed the groups consists of pro-C*-algebras with some Hopf-like structure. For Moore groups, these algebras are Hopf algebras with respect to the maximal C*-tensor product. We also show that this construction can be applied to some quantum groups. We expect that a similar construction, based on an envelope which takes into account all continuous representations, should cover all locally compact groups.

JULIA KUZNETSOVA (Université du Luxembourg)


25 October 2010 (Exceptional time: 14:15.)

COMBINATORIAL ASPECTS OF QUANTUM PERMUTATION GROUPS

The free analogue of the symmetric group Sn is the quantum group Sn+, introduced by Wang about 10 years ago. Quite surprisingly, this quantum group is infinite, starting from n=4. I will present a few combinatorial techniques, partly coming from compact groups and/or probability theory, which allow one to study this quantum group and some of its subgroups. This is based on joint work with Bichon, Collins, Curran, Skalski, Speicher, Vergnioux.

TEODOR BANICA (Université de Cergy-Pontoise, France)


8 November 2010

ON SOME CONVOLUTION EQUATIONS FOR STATES ON (LOCALLY) COMPACT QUANTUM GROUPS

The set of states on a (locally) compact quantum group equipped with the natural convolution multiplication forms a semigroup generalising the classical semigroup of probability measures on a group. Certain convolution equations have natural probabilistic or group-theoretic interpretations. For instance, the existence of the Haar state is equivalent to the existence of a (unique) state h such that h*f=f*h = h for all states f. In this talk, I will discuss idempotent states, i.e., solutions of the equation f*f=f, and describe conditions preventing the existence of non-trivial square roots of Haar states (that is non-trivial solutions of the equation f*f=h). The talk is based on the joint work with Uwe Franz, Reiji Tomatsu and Pekka Salmi.

ADAM SKALSKI (IMPAN)


29 November 2010

NONCOMMUTATIVE GEOMETRY WITH A TWIST

Several examples of noncommutative geometries with twisted trace have appeared recently. These include one-dimensional examples arising from KMS states and a two dimensional example for the standard Podleś sphere. I will describe ongoing work with Roger Senior and Ulrich Krähmer (SUq(2)), Jens Kaad and Andrzej Sitarz (an abstract framework), and Joseph C. Várilly (SUq(n)). All of these different lines of enquiry are aiming to get a clearer picture of what kind of `unbounded gadget' one needs to obtain a version of the local index formula when there is a twisted trace.

ADAM RENNIE (Australian National University, Canberra)


29 November 2010 (Exceptional time: 14:15.)

NONCOMMUTATIVE GEOMETRY BEYOND DEFORMATIONS OF COMMUTATIVE GEOMETRY

With each splitting of a polynomial into linear factors in polynomials with coefficients in an associative algebra, we associate a distributive law between monads. We prove that any splitting of a separable polynomial in the center of the algebra does not admit formal deformations with a non-trivial distributive law. Using this, we show an example of a noncommutative splitting that is not a deformation of any central splitting.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


20 December 2010

SEARCHING FOR CLASSICAL SUBSYSTEMS OF QUANTUM EVOLUTIONS

The notion of noncommutative topological entropy for automorphisms of (nuclear) C*-algebras was introduced in 1995 by D. Voiculescu as a generalisation of the topological entropy for continuous transformations of compact spaces. Most methods of computing the Voiculescu entropy are related to finding suitable commutative subsystems of noncommutative dynamical systems, which suggests a straightforward relation between the classical and quantum case. In this talk, we will explain some of the properties of the Voiculescu entropy and present recent examples related to bitstream shifts (studied by S. Neshveyev and E. Stormer) and to endomorphisms of Cuntz algebras. They show that the connections between the commutative and noncommutative case are actually quite subtle. Then, we will discuss the general problem of finding commutative subsystems of a given quantum dynamical system. Parts of the talk are based on the joint work with Jeong Hee Hong, Wojciech Szymański and Joachim Zacharias.

ADAM SKALSKI (IMPAN)


20 December 2010 (Exceptional time: 14:15.)

QUANTUM ALGEBRAIC SETS

Noncommutative geometry is a metaphore based on a duality between commutative algebras and spaces, i.e., sets equipped with an appropriate geometric structure. Under this duality, the elements of commutative algebra define functions on the corresponding set via the Gelfand transform, which is nothing but a (co)unit of the trivial (contravariant) adjunction between commutative algebras and sets. We construct a strong monoidal faithful and full embedding of the Cartesian category of sets into an appropriate monoidal category of "quantum sets", and extend the above adjunction to the adjunction between (associative) algebras and quantum sets. The image of algebras under this adjunction is called "quantum algebraic sets". Finally, we justify our construction by showing that any bialgebra is a coordinate algebra of a "quantum monoid", that is, a monoid in the monoidal category of quantum sets.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


3 January 2011

QUANTUM HOMOGENEOUS SPACES

The aim of this talk is to present and analyze a definition of a quantum homogeneous space of a locally compact quantum group. It was motivated by the Vaes construction of the quotient of a quantum group by a closed quantum subgroup. In this talk, I will prove that our framework is compatible with the definition of a quantum homogeneous space of a compact quantum group due to Podleś. I shall also describe the main steps of a proof that the Rieffel deformation of a homogeneous space is a quantum homogeneous space in our sense.

PAWEŁ Ł. KASPRZAK (Uniwersytet Warszawski)


10 January 2011

THE NONCOMMUTATIVE GEOMETRY OF INSTANTON MODULI SPACES

It has been known for some time now that the moduli space of self-dual gauge fields (instantons) on a compact four-manifold is an important invariant of its differential structure. It is therefore only natural to ask if the same is true for instantons on noncommutative four-manifolds. Indeed, it is of particular interest to ask if a given parameter space of instantons can 'detect' whether the underlying four-manifold is noncommutative or classical. In this seminar, I will describe some ongoing joint work with Giovanni Landi and Walter van Suijlekom, in which we try to give at least a partial answer to this question.

SIMON BRAIN (Université du Luxembourg)


10 January 2011 (Exceptional time: 14:15.)

REDUCTIONS OF PIECEWISE TRIVIAL PRINCIPAL COMODULE ALGEBRAS

The structure group of a principal bundle is reducible to a subgroup if there exists a local trivialisation with respect to which all transition functions take values in this subgroup. Conversely, if a principal bundle is reducible to a locally trivial principal sub-bundle, then there exists a local trivialisation of the bundle such that all transition functions take values in the structure group of the sub-bundle. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise trivial principal comodule algebra as a suitable replacement of a locally trivial compact principal bundle. To enclose natural and geometrically interesting noncommutative examples, we use smash products (cocycle-free crossed products) rather than tensor products as a generalisation of trivial principal bundles. These examples serve as a testing ground for our reduction theorem. (Joint work with P.M. Hajac.)

BARTOSZ P. ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


28 February 2011

BUNDLES OVER THE QUANTUM REAL PROJECTIVE SPACE

There are many non-trivial reductions of trivial principal bundles: if G' is a closed subgroup of G, then G -> G/G' prolongated from G' to G yields an Ehresmann groupoid which can be trivialised as a G-bundle by the group multiplication. Hence it is always a reduction of a natural trivial principal bundle. More generally, if X' is a principal G'-bundle admitting a G'-equivariant map from X' to G, the bundle X' is a reduction of the trivial bundle X'/G' x G. A natural embedding of the 2-sphere in SU(2) is Z/2-equivariant, and thus yields a reduction of the real projective space times SU(2) into the sphere Z/2-bundle. The aim of this talk is to show how a Z/2-equivariant embedding of the Podleś equator sphere in a quantum SU(2) gives a reduction of a smash product to the non-trivial bundle defining the quantum real projective space as a Z/2-quotient of the Podleś equator sphere. (Joint work with T. Brzeziński and P.M. Hajac.)

BARTOSZ P. ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


7 March 2011

AMENABILITY AND PROPERTY (T) FOR QUANTUM GROUPS

The notion of amenability was one of the first classical group-theoretic notions that was generalized to the quantum group setting, and it is by now fairly well understood in this general context. A somewhat opposite notion is the one of property (T): it has been intensively studied for groups but the corresponding notion for quantum groups is still in a very immature state. In my talk, I will give an introduction to both amenability and property (T) for discrete groups and quantum groups, and show how one can use these properties to construct new quantum-group completions of the Hopf *-algebra associated with a compact quantum group. This is joint work with Piotr M. Sołtan.

DAVID KYED (Universität Göttingen, Germany)


14 March 2011

BEYOND ELLIPTICITY

K-homology is the dual theory to K-theory. The Baum-Douglas (BD) isomorphism of topological and analytic K-homology can be taken as providing a framework within which the Atiyah-Singer index theorem can be extended to certain non-elliptic operators. This talk considers a class of non-elliptic differential operators that arise on compact contact manifolds. These operators have been studied by a number of mathematicians. Working within the BD framework, the index problem will be solved for these operators. This is joint work with Erik van Erp.

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


21 March 2011

THE DYNAMICAL QUANTUM GROUP SUq(2) ON THE LEVEL OF OPERATOR ALGEBRAS

Dynamical quantum groups were introduced by Etingov and Varchenko as an algebraic framework for the study of the quantum dynamical Yang-Baxter equation. They fit into the theory of Hopf algebroids developed by Böhm and others, and form a particular class of quantum groupoids. The simplest example of a dynamical quantum group is a variant of the compact quantum group SUq(2) of Woronowicz. On the algebraic level, its representation theory and relations to special functions were studied by Koelink and Rosengren. In this talk, I will start with some introduction to dynamical quantum groups, and then focus on the variant of SUq(2) associating to it a measured quantum groupoid in the sense of Enock and Lesieur and a proper reduced C*-quantum groupoid.

THOMAS TIMMERMANN (Universität Münster, Germany)


28 March 2011

RESTRICTING THE BI-EQUIVARIANT SPECTRAL TRIPLE ON QUANTUM SU(2) TO THE PODLEŚ SPHERES

It is known that the classical Dirac operators on the n-spheres can be constructed by an induction process from lower to higher dimensions. The question arises if a similar process is possible for spectral triples of noncommutative geometry. This time, since we work on the function level, the induction would be from higher to lower dimensions. The talk approaches this question by studying the relation between the bi-equivariant Dirac operator on quantum SU(2) and the spectral triples on the 1-parameter family of Podleś spheres. It turns out that the equatorial Podleś sphere is distinguished by being the unique one for which the restricted Dirac operator admits an equivariant grading operator. In the other cases, the restricted spectral triples are unitarily equivalent to the known equivariant even real spectral triples, but the unitary operators implementing the equivalence do not commute with the grading operator. Finally, the talk will end by discussing how the equivariant real structure is related to the Tomita operator on quantum SU(2) and the R-matrix operator of Uq(sl(2)).

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


4 April 2011 (Banach Center research group.)

LINE BUNDLES AND THE THOM CONSTRUCTION IN NONCOMMUTATIVE GEOMETRY

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we can construct Z and N-graded algebras, the Z-graded algebra being a Hopf-Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the N-graded algebra. In this case, we can carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu. (Joint work with T. Brzeziński.)

EDWIN J. BEGGS (Swansea University, Wales)


4 April 2011 (Exceptional time: 14:15. Banach Center research group.)

FIBRE-PRODUCT APPROACH TO INDEX PAIRINGS FOR THE STANDARD HOPF FIBRATION OF SUq(2)

The class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. In particular, this allows us to handle cases of C*-algebras lacking two different non-trivial ideals, such as the C*-algebra of the standard Podleś sphere. The goal of this talk is to present an index computation for noncommutative line bundles over the standard Podleś sphere using the Mayer-Vietoris type arguments afforded by a one-surjective pullback presentation of the C*-algebra of this quantum sphere.

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


18 April 2011

THE KLEIN-PODLEŚ BOTTLE AS A NON-TRIVIAL BUNDLE OVER THE QUANTUM REAL PROJECTIVE SPACE RPq(2)

We tensor the C*-algebra of the equatorial Podleś quantum sphere with the algebra of continuous functions on the unit circle, act on the tensor product with the diagonal antipodal Z/2-action, and consider the invariant subalgebra. This gives a U(1)-C*-algebra A with the quantum real projective space C*-algebra C(RPq(2)) as its U(1)-invariant part. Using the identity representation of U(1), we associate with it a finitely generated projective module over C(RPq(2)). Combining methods of topology and operator algebras, we prove that this module is not free. This implies that A cannot be a crossed product of C(RPq(2)) and the integers. To compute the K-theory of A, we present it as a pullback of two copies of the tensor product of the equatorial sphere C*-algebra with the algebra of continuous functions on the interval [0,1]. We put these two copies together by the identity and antipodal automorphisms of the quantum sphere applied at 0 and 1, respectively. Hence A can be viewed as defining a non-trivial bundle over a circle with the fibre being the equatorial quantum sphere. Since replacing the quantum sphere with the unit circle and the antipodal action with the complex conjugation would yield the Klein bottle, we call A the C*-algebra of the Klein-Podleś bottle. (Joint work with P.F. Baum.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)


9 May 2011

EQUIVARIANT MORITA EQUIVALENCES BETWEEN PODLEŚ SPHERES

Recently, I noticed that the Podleś spheres can be realized as subquotients of the quantum universal enveloping algebra of su(2) with a twisted *-operation. This observation allows one to show in an efficient way that there exists an equivariant Morita equivalence between two Podleś spheres if and only if there is an integral relationship between their parameters. It also leads to an explicit description of the actions that are equivariantly Morita equivalent with the action on the quantum projective plane. In this seminar, I will explain my approach to proving the above results paying special attention to the `borderline friction' between the algebraic and the analytic aspects of the subject.

KENNY DE COMMER (University of Roma Tor Vergata / CMTP, Italy)


9 May 2011 (Exceptional time: 14:15.)

ON THE CONSTRUCTION OF QUANTUM HOMOGENEOUS SPACES FROM *-GALOIS OBJECTS

I will explain how to construct bi-*-Galois objects that allow one to pass from compact to non-compact forms of the quantized universal enveloping algebras of simple complex Lie algebras. I will also show how one can create quantum homogeneous spaces for the associated quantum groups by integrating the Miyashita-Ulbrich action on appropriate subquotient *-algebras. Finally, I will discuss how this might lead to operator algebraic versions of the quantizations of some non-compact Lie groups. (Work in progress.)

KENNY DE COMMER (University of Roma Tor Vergata / CMTP, Italy)


23 May 2011 (Banach Center research group.)

CYCLIC STRUCTURES IN ALGEBRAIC (CO)HOMOLOGY THEORIES

The cyclic cohomology of a left Hopf algebroid with coefficients in a right module left comodule is defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. A form of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti-Yetter-Drinfeld modules. (Joint work with N. Kowalzig.)

ULRICH KRÄHMER (University of Glasgow, Scotland)


23 May 2010 (Exceptional time: 14:15. Banach Center research group.)

CUP PRODUCTS IN HOPF-CYCLIC COHOMOLOGY WITH COEFFICIENTS IN CONTRAMODULES

We use stable anti-Yetter-Drinfeld contramodules as coefficients of Hopf-cyclic cohomology to achieve the functoriality of cup products. This generalization of original cup products in Hopf-cyclic cohomology allows us to define them for two different stable anti-Yetter-Drinfeld modules. To show that the choice of coefficients is important, we prove the non-trivial dependence of cup products on coefficients.

BAHRAM RANGIPOUR (University of New Brunswick, Fredericton, Canada)


30 May 2011

EQUIVARIANT HOPF-GALOIS (CO)EXTENSIONS AND HOPF-CYCLIC COHOMOLOGY

We define the notions of a cross Hopf coalgebra and an equivariant Hopf-Galois (co)extension. We show that (co)extensions yield functors between the categories of stable anti-Yetter-Drinfeld modules over the cross Hopf (co)algebras involved in the (co)extension. This way we generalize earlier constructions associating a stable anti-Yetter-Drinfeld module to a Hopf-Galois extension. (Joint work with M. Hassanzadeh.)

BAHRAM RANGIPOUR (University of New Brunswick, Fredericton, Canada)


30 May 2011 (Exceptional time: 14:15.)

COMPLETE CLASSIFICATION OF STABLE ANTI-YETTER-DRINFELD MODULES OVER BICROSSED PRODUCT HOPF ALGEBRAS

To complete our earlier study of stable anti-Yetter-Drinfeld modules over bicrossed product Hopf algebras, we define such modules over Lie algebras. We show that the Weil algebra of a Lie algebra gives a natural example of the Hopf-cyclic cohomology of Lie algebras with coefficients in a stable anti-Yetter-Drinfeld module. We provide a one-to-one correspondence between the stable anti-Yetter-Drinfeld modules over the bicrossed product Hopf algebra associated to a matched pair of Lie algebras and the stable anti-Yetter-Drinfeld modules over the total Lie algebra of this matched pair. Using this correspondence, we show that the only finite-dimensional stable anti-Yetter-Drinfeld modules over the Connes-Moscovici Hopf algebras are the one-dimensional examples found in 1998. (Joint work with Serkan Sutlu.)

BAHRAM RANGIPOUR (University of New Brunswick, Fredericton, Canada)