NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Sniadeckich 8, room 408, Mondays, 10:15-12:00

1999/2001 2002/2003 2003/2004 2004/2005

3 October 2005

BRAIDED CYCLIC COHOMOLOGY

I will give an introduction to the braided cyclic cohomology of Majid and Akrami. This is developed in the general setting of so-called ribbon algebras in braided monoidal categories. In the trivially braided case, braided cyclic cohomology reduces to the twisted cyclic cohomology of Kustermans, Murphy and Tuset. An important related notion is that of a cocycle twist of a Hopf algebra, previously studied by Majid and Oeckl, who showed that bicovariant differential calculi are preserved under twisting. Majid and Akrami proved that braided cyclic cohomology is invariant under cocycle twists. This will be an introductory talk and all terminology will be explained.

TOM HADFIELD (Queen Mary, University of London, England)

10 October 2005

ON THE HOCHSCHILD HOMOLOGY OF QUANTUM SL(N)

It is shown that the standard quantized coordinate ring A:=k_q[SL(N)] satisfies Van den Bergh's analogue of the Poincare duality for Hochschild (co)homology. Here the dualizing bimodule is A_\sigma, ie the bimodule that is A as a left A-module but with the right multiplication twisted by the modular automorphism \sigma of the Haar functional. This implies, in particular, H_{N^2-1}(A,A_\sigma)=k.

ULRICH KRAEHMER (Instytut Matematyczny, Polska Akademia Nauk)

17 October 2005

VECTOR BUNDLES AND PROJECTIVE MODULES

In 1955, J.P. Serre showed that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring. Later, in 1961, G. Swan showed analogous correspondence for topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C(X) of continuous functions on X (real, complex or quaternion - valued depending on the type of vector bundles under consideration). This result can be extended to the paracompact Hausdorff spaces provided we will restrict ourselves to the bundles of finite type. In the talk I will present the proof of the most general result due to L. Vaserstein (Vector bundles and projective modules), which holds for an arbitrary topological space X. (In particular, it does not need to be Hausdorff.) Of course, one has to give the correct definition of the bundles of the finite type in the situation of arbitrary topological spaces. The result can be formulated as an equivalence of categories of finitely generated projective modules over C(X) and the category of vector bundles of finite type over X. In analogy with the well-known case of compact Hausdorff spaces, there is also a homotopical classification of vector bundles of finite type over general spaces, which will be sketched after L. Vaserstein.

PAWEL WITKOWSKI (Instytut Matematyki, Uniwersytet Warszawski)

24 October 2005

K-THEORETIC DUALITY FOR HIGHER RANK GRAPH ALGEBRAS AND BOUNDARY CROSSED PRODUCTS

Classical Poincare duality says that there is a canonical isomorphism between the cohomology and homology of a compact manifold implemented by the cap-product with a certain fundamental class. Kasparov's KK-theory allows to formulate an analogue of Poincare duality for noncommutative C*-algebras. We establish such a duality result for a class of higher rank graph algebras and discuss applications to duality for groups acting on buildings. (Based on the joint work with Iulian Popescu.)

JOACHIM ZACHARIAS (The University of Nottingham, England)

7 November 2005

FUNDAMENTAL PROBLEMS OF NONCOMMUTATIVE TOPOLOGY

The Gelfand-Naimark duality between the category of commutative unital C*-algebras and the category of compact Hausdorff spaces leads to various noncommutative eneralizations of topological constructions. Also, it inspires interpretation of some constructions with noncommutative C*-algebras in a `dual topological' language. This approach follows the following recipe. First, take a property (construction, theorem etc.) of compact Hausdorff spaces and dualize it to the category of commutative unital C*-algebras. Then generalize it to noncommutative unital C*-algebras, and finally dualize it to the opposite category of the category of (noncommutative) unital C*-algebras. The latter category is viewed as the category of `noncommutative compact Hausdorff spaces'. One interpretes results obtained in the aforementioned way as a `noncommutative generalization' of the initial topological property (construction, theorem etc.). A pedagogical panorama of results and problems in this field will be presented.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)

14 November 2005

THE TOMITA-TAKESAKI CONSTRUCTION - A FAMOUS REVOLUTION IN THE THEORY OF OPERATOR ALGEBRAS

When Tomita-Takesaki theory was discovered in late 60's, for operator algebraists it was like a true revolution. It gave at their disposal completely new and unexpected tools. I will try to explain the basic construction of Tomita-Takesaki theory and its significance for statistical physics (KMS states) and for the classification of factors (Connes' invariants).

JAN DEREZINSKI (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)

21 November 2005

THE RELATIVE CHERN-GALOIS CHARACTER

The Chern-Galois theory is developed for corings or coalgebras over non-commutative rings. As the first step the notion of an entwined extension as an extension of algebras within a bijective entwining structure over a non-commutative ring is introduced. A strong connection for an entwined extension is defined and it is shown to be closely related to the Galois property and to the equivariant projectivity of the extension. A generalisation of the Doi theorem on total integrals in the framework of entwining structures over a non-commutative ring is obtained, and the bearing of strong connections on properties such as faithful flatness or relative injectivity is revealed. A family of morphisms between the K0-group of the category of finitely generated projective comodules of a coring and even relative cyclic homology groups of the base algebra of an entwined extension with a strong connection is constructed. This is termed a relative Chern-Galois character. Explicit examples include the computation of a Chern-Galois character of depth 2 Frobenius split (or separable) extensions over a separable algebra R. Finitely generated and projective modules are associated to an entwined extension with a strong connection, the explicit form of idempotents is derived, the corresponding (relative) Chern characters are computed, and their connection with the relative Chern-Galois character is explained. Joint work with Gabriella Bohm (Budapest).

TOMASZ BRZEZINSKI (University of Wales, Swansea, Wales)

28 November 2005

NONCOMMUTATIVE TOPOLOGY REVISITED

We will discuss the problem of comparison of classical (commutative) topology with constructions on the level of noncommutative algebras. The difficulty lies in the impossibility of defining an open subset as an object in the category of Gelfand transforms of commutative C*-algebras. The concept of a locale and a quantale as a noncommutative replacement for the lattice of open subsets will be proposed. As an exercise in dealing with open subsets, we will prove the easy direction of the Tietze-Urysohn extension theorem (if functions can be extended, the space is normal) for arbitrary C*-algebras.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)

5 December 2005

THE NONCOMMUTATIVE GEOMETRY OF GRAPH C*-ALGEBRAS

Graph C*algebras and k-graph C*-algebras provide an extremely large and diverse class of C*-algebras. They have been widely studied in part because they are amenable to computation. In this talk I will describe joint work with David Pask and Aidan Sims which shows that for a large class of these algebras, we can construct spectral triples and compute indices. The novel feature of these examples is that they are naturally semifinite-meaning that we do not use the operator trace on B(H), but a different semifinite trace. The trace we use arises naturally from the geometry of the graph. The resulting semifinite index is an invariant of a finer structure than the isomorphism class of the algebra.

ADAM C. RENNIE (University of Copenhagen, Denmark)

6 December 2005 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5810, 12:00 Tuesday.)

COMMUTATIVE GEOMETRIES ARE SPIN MANIFOLDS

I will describe the tools and approach that Joe Varilly and myself employ to reconstruct a manifold from a spectral triple. This will include some discussion of the axioms/conditions employed, as well the role they play in the reconstruction.

ADAM C. RENNIE (University of Copenhagen, Denmark)

12 December 2005

PROPER ACTIONS

Different definitions of a proper action will be discussed (Bourbaki, Palais, Baum-Connes-Higson). The conditions on a topological space which make them equivalent will be given. The talk will be based on the work of Harald Biller "Characterizations of Proper Actions".

PIOTR STACHURA (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)

19 December 2005

LOCALLY COALGEBRA-GALOIS EXTENSIONS

The notion of a locally coalgebra-Galois extension is introduced. As a special case, we discuss the concept of a locally cleft extension. The necessary and sufficient conditions are given for a locally coalgebra-Galois extension to be a (global) coalgebra-Galois extension. As examples, we consider the quantum lens space constructed by gluing two quantum solid tori.

BARTOSZ ZIELINSKI (Uniwersytet Lodzki)

9 January 2006

ARE COMMUTATIVE GEOMETRIES SPIN MANIFOLDS?

This is a sequel to the talk by Adam Rennie with a similar title. Starting from a spectral triple over a commutative *-algebra, I shall discuss how the axioms/conditions which we employ allow us to reconstruct a certain vector bundle, and then to build up the manifold of which it is the cotangent bundle.

JOSEPH C. VARILLY (University of Costa Rica, Costa Rica)

16 January 2006

HEISENBERG-LORENTZ QUANTUM GROUP

At the first part of the seminar, I will describe a new approach to the Rieffel deformation of C*-algebras. It is based on the twisting (by means of a dual 2-cocycle) of the dual action in the crossed product construction. Taking the Landstad invariants of this action, we obtain a deformed C*-algebra. In the case of R^n actions, we reproduce the aforementioned Rieffel deformations. This approach allows us to obtain a very short proof of the invariance of K-theory under the Rieffel deformations. Next, I will apply this to SL(2,C) endowed with a chosen abelian subgroup. This will lead to a quantum group. I will show that it is generated (in the sense of Woronowicz) by four unbounded elements corresponding to the standard coordinate functions on SL(2,C). They satisfy interesting commutation relations, which originate the name Heisenberg-Lorentz quantum group.

PAWEL L. KASPRZAK (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)

23 January 2005

PAIRING IN HOPF-CYCLIC COHOMOLOGY AND BIVARIANT THEORY

Using the Cuntz-Quillen formalism of towers and X-complexes, we define a map from the (coalgebra) cyclic cohomology of a coalgebra C, acting on an algebra A, to the bivariant cyclic cohomology of A. A similar map can be constructed in the presence of a Hopf algebra H acting on both C and A in a coherent way, and a stable anti-Yetter-Drinfeld H-module M of coefficients. In this case, we obtain a map from the Hopf-cyclic cohomology of C (with coefficients in M) to the bivariant cyclic cohomology with the first entry A and the second entry constructed from A, H and M. Pairing the image of this map with Hopf-cyclic cohomology classes in of A, we obtain a cup product similar to the one introduced by M.Khalkhali and B.Rangipour. We conjecture that the latter coincides with our pairing. This is supported by the fact that they coincide on the level of periodic cohomology. Our construction is a far-reaching generalization of a construction by M.Crainic. One of its easy consequences is the possibility to obtain a cap-product of the (usual) cyclic homology of A and the Hopf-cyclic cohomology of C with values in the Hopf-cyclic homology of A.

GEORGIY SHARYGIN (ITEP, Moscow, Russia)

20 February 2006

EXCISION IN HOPF-CYCLIC COHOMOLOGY

This talk is a review of the excision problem in Hochschild and cyclic homology, ranging from the classical results of Wodzicki to the latest developments for Hopf cyclic homology due to Kaygun and Khalkhali.

ULRICH KRAEHMER (Instytut Matematyczny, Polska Akademia Nauk)

27 February 2006

MAXIMAL COMMUTATIVE SUBALGEBRAS AND HOCHSCHILD HOMOLOGY

A spectral sequence convergent to Hochschild homology with coefficients in a bimodule will be presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its second term is computed in terms of Poisson homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.

TOMASZ MASZCZYK (Instytut Matematyczny PAN, Instytut Matematyki UW)

6 March 2006

ON THE CONSTRUCTION OF INSTANTONS IN NONCOMMUTATIVE GEOMETRY

We consider three types of noncommutative deformations of the SU(2)-principal bundle of the 7-dimensional sphere over the 4-dimensional sphere. These are the coisotropic deformation (inspired by Poisson geometry and based on the Soibelman-Vaksman 7-sphere), the isospectral deformation, and the deformation based on symplectic quantum groups. In the latter two cases, a projection giving the basic instanton of unit charge will be constructed. The problem to generalize this picture to arbitrary SU(2)-instantons will be discussed at the end of the talk.

CHIARA PAGANI (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)

13 March 2006

NON-CROSSED-PRODUCT EXAMPLES OF PRINCIPAL EXTENSIONS OF C*-ALGEBRAS

The aim of this talk is to present a method for proving that a given C*-algebra equipped with a free action of U(1) is not a crossed product with its fixed-point subalgebra. It is based on the standard Fourier analysis with coefficients in a C*-algebra, and the computation of an appropriate K-invariant (Fredholm index) of a finitely generated projective module associated to the U(1)-action. This is in analogy with the classical fact that, if a principal bundle admits an associated vector bundle which is not trivial, then it itself cannot be trivial. (Based on a joint work with R.Matthes and W.Szymanski.)

PIOTR M. HAJAC (Instytut Matematyczny PAN / Katedra Metod Matematycznych Fizyki UW)

20 March 2006

NONCOMMUTATIVE LENS SPACES

In this talk, we examine two families of "3-dimensional" noncommutative lens spaces as examples of C*-algebras equipped with free actions of U(1). To prove that they are not crossed products with their fixed-point subalgebras, we carry out appropriate index computation for associated projective modules. These two families are related to the Heegaard-type noncommutative 3-sphere and the quantum SU(2), respectively. (Based on joint research with P.M.Hajac, J.H.Hong and R.Matthes. See Heegaard-type and SU(2)-type (part1), SU(2)-type (part2), for details.)

WOJCIECH SZYMANSKI (The University of Newcastle, Australia)

27 March 2006

ACTIONS AND SUBGROUPS OF COMPACT QUANTUM GROUPS

I will describe basic notions related to an action of a compact quantum group. This will include quantum subgroups and different types of noncommutative homogeneous spaces. Time permitting, I shall discuss the classification of subgroups of the quantum SU(2) group. (This talk will be essentially based on a work of Piotr Podles.)

PIOTR M. SOLTAN (Katedra Metod Matematycznych Fizyki UW)

3 April 2006

LOCAL INDEX FORMULA ON SUq(2)

The local index formula of Connes-Moscovici for a recently constructed isospectral Dirac operator on the quantum SU(2) will be presented. The `cosphere bundle' and the dimension spectrum {1,2,3} coincide with the one obtained by Connes for the spectral triple of Chakraborty and Pal, but the local cyclic cocycles yielding the index formula turn out to be slightly different. (Based on a joint work with W. van Suijlekom, G. Landi, A. Sitarz and J.C. Varilly.)

LUDWIK DABROWSKI (SISSA, Trieste, Italy)

24 April 2006

HIGHER DIMENSIONAL MIRROR QUANTUM SPHERES

In this talk, we present a construction of noncommutative "n-dimensional" spheres by glueing two copies of noncommutative discs. Our method extends the construction of the "2-dimensional" mirror quantum sphere of Hajac, Matthes and Szymanski. The noncommutative discs are obained through repeated application of a quantum double suspension. Both C*-algebras and the polynomial algebras of the discs and spheres are discussed. Our construction yields new examples of noncommutative spheres in all even dimensions. (Based on ongoing joint research project with J.H.Hong.)

WOJCIECH SZYMANSKI (The University of Newcastle, Australia)

8 May 2006

WHAT IS AN EQUIVARIANT INDEX? (full contents in pdf)

Let G be a (countable) discrete group. Assume that G is acting by a smooth proper and co-compact action on a manifold M. Let D be a G-equivariant elliptic operator on M. What do we mean by the G-index of D? This talk proposes an answer to this question. Most of the talk is devoted to one example. In this example, the discrete group is the integers Z, the manifold is the real line R (acted on by Z in the usual way so that the quotient space is the circle), and the elliptic operator is -id/dx. Application: elliptic operator formulation of the Baum-Connes conjecture. Some (quite elementary) operator theory will be used in this talk.

PAUL F. BAUM (Penn State University, USA)

9 May 2006 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5810, 12:00 Tuesday.)

EQUIVARIANT CHERN CHARACTER (full contents in pdf)

Let G be a (countable) discrete group acting by a smooth action on a manifold M. There is no further hypothesis on the action. C*(G,M) denotes the reduced crossed-product C*-algebra arising from the action of G on M. If G is finite, then the K-theory of C*(G,M) is Atiyah-Segal equivariant K-theory. When G is not finite, the K-theory of C*(G,M) can be viewed as the natural generalization of Atiyah-Segal equivariant K-theory. What should be the target of the Chern character whose source is the K-theory of C*(G,M)? In this talk, the target is defined in terms of classical homological algebra. Two extreme cases are then examined: the case when the action of G on M is proper - and the case when the manifold M is a point.

PAUL F. BAUM (Penn State University, USA)

15 May 2006

SPIN STRUCTURES IN NONCOMMUTATIVE GEOMETRY

In differential geometry it is well known that the spectrum of the Dirac operator may depend on the choice of a spin structure. We show that a similar effect happens in Noncommutative Geometry. In the case study of the noncommutative torus, we prove the existence of inequivalent equivariant real spectral geometries that in the classical limit correspond to the different choices of a spin structure. We construct also spectral triples (for quantum spheres) that have no classical counterparts and lead to quasi-Dirac operators on the sphere. We present local index computations in these situations.

ANDRZEJ SITARZ (Universitaet Duesseldorf, Germany / Uniwersytet Jagiellonski, Poland)

22 May 2006

ACTIONS OF COMPACT QUANTUM GROUPS ON C*-ALGEBRAS

In the first part of this talk, we investigate the structure of the fixed-point algebra under an action of a compact matrix quantum group on a C*-algebra B. We also show that the categories of comodules in B and inner endomorphisms restricted to the fixed-point algebra coincide when the relative commutant of the fixed-point algebra is trivial. Next, we show a version of the Tannaka duality theorem for the unitary quantum groups SUq(N) of Woronowicz. The second part will be devoted to ergodic actions of compact quantum groups on unital C*-algebras. In particular, we show that the unique invariant state has special modular properties.

MARCIN MARCINIAK (Uniwersytet Gdanski, Poland)

29 May 2006

QUANTUM DUALITY PRINCIPLE FOR q-HOMOGENEOUS SPACES AND APPLICATIONS

Drinfeld's quantum duality principle states that quantized universal enveloping algebras may be understood as quantization of function algebras on the dual Poisson-Lie group. Once the concept of a quantum coisotropic subgroup is taken into account, we will show how this principle extends to the context of quantum subgroups and quantum homogeneous spaces. This leads us to consider the natural notion of complementary dual Poisson and quantum homogeneous spaces. We will apply this principle to produce new examples of quantum homogeneous spaces from known ones. Time permitting, we will show that this applies to a family of quantum (Poisson) complex Grassmannians. (Based on joint work with F.Gavarini.)

NICOLA CICCOLI (Universita di Perugia, Italy)

5 June 2006

TWISTED HOCHSCHILD (CO)HOMOLOGY FOR NOETHERIAN HOPF ALGEBRAS

I will discuss the recent paper Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras by K.A. Brown and J.J. Zhang. The motivation for this work is as follows. In the last twenty years, many interesting noncommutative algebras arising from quantum groups have appeared and been intensively studied. Early on, it was noticed that in many situations where noncommutative algebras are obtained via deformations of appropriate commutative function algebras on spaces, the Hochschild dimension of the deformed algebra is strictly less than that of the commutative algebra one started with - homological information has been lost, and this "dimension drop" phenomenon was interpreted by some to indicate that Connes' noncommutative geometry was ill-suited to quantum groups. In joint work with U. Kraehmer, motivated by the twisted cyclic cohomology of Kustermans, Murphy and Tuset, we showed first for quantum SL(2) and subsequently for quantum SL(N) that the "dimension drop" can be overcome by passing to coefficients in an appropriate "twisted" bimodule - with twisting coming from the canonical modular automorphism associated to the Haar functional on the quantum group. Similar results were obtained for Podles quantum spheres (by myself) and quantum hyperplanes (A. Sitarz). In recent work Brown and Zhang have shown that similar results apply for all known classes of Noetherian Hopf algebras - specifically, that for any such algebra the global dimension can be reached as Hochschild dimension with coefficients in an appropriate bimodule twisted by the so-called Nakayama automorphism. In particular their results apply to quantised enveloping algebras and quantised function algebras of semisimple groups. In this talk we will give an introduction to their work and discuss its implications.

TOM HADFIELD (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)