NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

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


2 October 2006

PRINCIPAL HOPF-ALGEBRA EXTENSIONS FROM THE YETTER-DRINFELD BRAIDING

A simple topological construction allows one to produce non-trivial principal bundles for all non-contractible compact groups. (This includes all non-trivial compact Lie groups.) For instance, this way one can obtain the edge of the Moebius strip over the circle, the Hopf fibration, and the instanton fibration. The aim of this talk is to show how to carry out this construction for an arbitrary Hopf *-algebra H. A key step is to use an appropriate braiding on the tensor square of H, so as to make the diagonal coaction of H a *-algebra homomorphism. The main result is a general and explicit formula for a strong connection. The existence of a strong connection puts us into the framework of principal extensions and K-theory, and its explicit form gives us concrete idempotents representing associated finitely generated projective modules. (Based on a joint work with L.Dabrowski and T.Hadfield.)

PIOTR M. HAJAC (Instytut Matematyczny, Polska Akademia Nauk / Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)


9 October 2006

TWISTED SPECTRAL TRIPLES AND COVARIANT DIFFERENTIAL CALCULI OVER QUANTUM SU(2)

Twisted spectral triples generalise the standard notion of spectral triples by replacing commutators [D,a] by Da - s(a)D, where s is an automorphism of the algebra under consideration. The aim of this talk is to point out that this concept (introduced by Connes and Moscovici in math.OA/0609703) yields a Hilbert space representation of precisely one of Heckenberger's 3-dimensional covariant differential calculi on quantum SU(2).

ULRICH KRAEHMER (Instytut Matematyczny, Polska Akademia Nauk)


16 October 2006

NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves, we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and alternative approaches using (in general non-monoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes.

TOMASZ MASZCZYK (Instytut Matematyki UW / Instytut Matematyczny PAN)


23 October 2006

BEYOND HOPF ALGEBRAS

In the theory of Hopf algebras there is a structure theorem which is very useful. It says that, in characteristic zero, a connected cocommutative Hopf algebra is cofree as a coalgebra and is, as an algebra, isomorphic to the universal enveloping algebra of a Lie algebra. This structure theorem is essentially equivalent to the union of the Poincare-Birkhoff-Witt theorem with the Cartier-Milnor-Moore theorem. It involves three types of algebras, that is three operads: Com for the coalgebra structure, As for the algebra structure, and Lie for the structure of the primitive part. The purpose of this talk is to show that there are numerous other examples of this form, many of them already in the literature. We give elementary conditions on a triple of operads (C, A, P), so that there is an analogous structure theorem. The paradigm is (Com, As, Lie). (See Generalized bialgebras and triples of operads for details.)

JEAN-LOUIS LODAY (CNRS, Strasbourg, France)


30 October 2006

BIVARIANT HOPF-CYCLIC COHOMOLOGY

We will define a bivariant version of Hopf-cyclic cohomology. We shall show that, when specialized to the ground field in either variables, it will give the ordinary Hopf-cyclic cohomology and dual Hopf-cyclic cohomology of module (co)algebras, respectively. (Joint work with Masoud Khalkhali.)

ATABEY KAYGUN (University of Western Ontario, London, Canada)


6 November 2006

DIRAC OPERATORS ON ALL PODLES QUANTUM SPHERES

A construction of equivariant spectral triples for all Podles quantum spheres is given. In particular cases, these spectral triples are regular, even and of metric dimension 2. The conditions on an equivariant real structure (commutant property and the first order condition) can be satisfied up to infinitesimals of arbitrary order.

ELMAR WAGNER (University of Trieste, Italy)


13 November 2006

MORITA INVARIANCE OF HOPF-CYCLIC (CO)HOMOLOGY

We will prove the Morita invariance of Hopf-cyclic (co)homology (with general coefficients) of module (co)algebras. Herein we understand the Morita equivalance as the equivalence of certain categories of (co)representations. (Joint work with Masoud Khalkhali.)

ATABEY KAYGUN (University of Western Ontario, London, Canada)


20 November 2006

X-COMPLEXES AND HOPF-CYCLIC (CO)HOMOLOGY

The notion of an X-complex was introduced by J.Cuntz and D.Quillen in a series of papers in the mid-90s. It proved to be a quite powerful instrument for treating cyclic (co)homology of algebras. In particular, it enables one to prove the excision in periodic cyclic cohomology. In this talk, I will explain their constructions and show how they should be modified if one wants to deal with Hopf-cyclic (co)homology with coefficients.

GEORGY SHARYGIN (ITEP, Moscow, Russia)


27 November 2006

MEGATRACE

All known reasonable constructions of exotic operator traces are specializations of one single construction: the `megatrace'.

MARIUSZ WODZICKI (University of California at Berkeley, USA)


4 December 2006

K-THEORETIC CONSTRUCTION OF NONCOMMUTATIVE INSTANTONS OF ARBITRARY CHARGES

We study noncommutative four-spheres constructed via gluings (fiber products of algebras) of noncommutative four-balls over their boundaries. Using Milnor's map from the K_1 of the three-sphere algebra to the K_0 of the four-sphere algebra, we obtain a family of finitely generated projective modules indexed by integers (the winding number). By the associativity of the Kasparov product underlying the boundary maps in the Mayer-Vietoris six-term exact sequence of KK-groups, the index of these projective modules can be computed via the index pairing of the defining unitaries over the three-sphere algebra. As an example we consider an explicit gluing over quantum SU(2). This corresponds to the instanton construction of Pflaum. We obtain explicit 4-by-4 idempotents representing all instanton bundles. Moreover, we prove that the index pairing of an instanton bundle with an appropriate Fredholm module (the charge of an instanton) always coincides with the winding number. (Joint work with L.Dabrowski, P.M.Hajac, R.Matthes.)

TOM HADFIELD (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)


11 December 2006

CUP PRODUCTS IN HOPF-CYCLIC COHOMOLOGY

In this talk, we will define cup products in Hopf-cyclic cohomology by using Yoneda composition and bivariant Hopf-cyclic cohomology. We will also show that Connes-Moscovici characteristic map in Hopf-cyclic cohomology fits into this framework.

ATABEY KAYGUN (University of Western Ontario, London, Canada)


8 January 2007

EULERIAN IDEMPOTENT, KASHIWARA-VERGNE CONJECTURE AND CYCLIC HOMOLOGY

By using the interplay of the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution (F,G) of the first equation of the Kashiwara-Vergne conjecture: x+y-log(exp(y)exp(x))=(1-exp(-ad x))F(x,y)+(exp(ad y)-1)G(x,y). Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent. In the last part, we show how to use the Eulerian idempotent to split cyclic homology.

EMILY BURGUNDER (Montpellier, France)


9 January 2007 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5810, 12:00 Tuesday.)

THE MULTIFACETTES OF THE STASHEFF POLYTOPE

The Stasheff complex is a polytope whose vertices are in bijection with the planar binary trees (one polytope for each n). It plays a role in many problems in algebraic topology (loop spaces, compactification). Recently it appeared also in algebraic combinatorics and in noncommutative geometry. In this lecture I will present a simple construction of these polytopes and show that the minimal triangulation is related to the ``parking functions".

JEAN-LOUIS LODAY (CNRS, Strasbourg, France)


15 January 2007

PIECEWISE TRIVIAL PRINCIPAL EXTENSIONS OF NONCOMMUTATIVE ALGEBRAS

To adapt the standard notion of local triviality of principal bundles to fibre products of C*-algebras, we introduce the concept of piecewise triviality. Using Paul Baum's example called the bubble space, we explain the key difference between the two definitions. Then we define and exemplify piecewise triviality for principal extensions of arbitrary unital algebras. This brings us to the main part of the talk: proving that every piecewise trivial extension is principal (locally trivial or piecewise trivial compact G-spaces are principal) and determining the piecewise structure of principal extensions of fibre-product algebras. (Based on a joint work with P.M.Hajac, U.Kraehmer and R.Matthes.)

BARTOSZ ZIELINSKI (IMPAN / Uniwersytet Lodzki)


22 January 2007 (Please note the room change: 403.)

NON-COMMUTATIVE DUALITIES

I will discuss some basic notions of spaces and sheaves in the framework of derived algebraic geometry, where one replaces usual associative algebras by differential graded algebras and, more generally, by triangulated differential graded categories. The ideal object is a smooth non-commutative space of finite type, for which examples are acyclic quivers with relations, smooth (commutative) algebraic varieties, free algebras, finite cell complexes, quantum projective spaces and algebraic quantum groups. Non-commutative definitions come in dual pairs:

perfect complex <-----> finite-dimensional representation
smoothness <-----> compactness
compactification <-----> resolution of singularities

Also, I will describe Serre dualities, Calabi-Yau properties, and a remarkable analogy between von Neumann algebras in functional analysis, and Koszul duality and formal completions in the derived setting.

MAXIM KONTSEVICH (IHES, Bures-sur-Yvette, France)


19 February 2007

TOWARDS AN ANALOGUE OF THE BAUM-CONNES CONJECTURE FOR DISCRETE QUANTUM GROUPS

We shall first review the approach to the Baum-Connes conjecture developped by Meyer and Nest using the language of triangulated categories and derived functors. Then we will describe some constructions needed to extend this machinery to the setting of quantum groups. As an illustration, we will explain how to formulate and prove the conjecture for the dual of the quantum SU(2) group.

CHRISTIAN VOIGT (Universitaet Muenster, Germany)


26 February 2007

QUANTUM GROUPS AND TWISTED SPECTRAL TRIPLES

In a recent paper, Connes and Moscovici explain how a twist in the original definition of a spectral triple makes it possible to deal with algebras with no (or few) traces. We review the basic properties of twisted spectral triples, their application to the construction of differential calculi and local index formulas, and compare the situation with the untwisted case. Then, we discuss how these notions fit into the framework of quantum homogeneous spaces, and illustrate what can be done in the example of the Landi-Pagani-Reina quantum symplectic 4-sphere.

FRANCESCO D'ANDREA (SISSA, Trieste, Italy)


5 March 2007

POINCARE DUALITY ON NONCOMMUTATIVE MANIFOLDS

In classical differential geometry, one of the main properties of a compact diffierentiable manifold M of dimension n is the Poincare duality, which establishes an isomorphism between cohomology in degree k and homology in degree n-k, or equivalently, it provides a non degenerate complex valued pairing between cohomology groups in degree k and n-k. An analogue of this property, expressed in terms of Poincare duality in Kasparov's KK-theory, has found a place in Connes' axiomatic description of differentiable manifolds in noncommutative geometry. In this talk we shall give an introduction to the notion of Poincare duality in bivariant K-theory and provide applications of this formalism in the D-brane theory. We propose a general formula for D-brane charge. We shall also discuss the Gysin map in this context and the Grothendieck-Riemann-Roch theorem.

JACEK BRODZKI (University of Southampton, G. Britain)


12 March 2007

HOCHSCHILD AND CYCLIC (CO)HOMOLOGY OF HOPF-GALOIS EXTENSIONS

Hochschild and cyclic cohomology are important invariants of algebras. In general, their computation is a very difficult task. In the case of Hopf-Galois extensions there are results that help to overcome this difficulty. The aim of my talk is to survey some of them. First, for a Hopf-Galois extension of B to A, there are several spectral sequences converging to the Hochschild (co)homology of A. I shall focus on the construction of a variant of Hochschild-Lyndon-Serre spectral sequence. Secondly, in certain cases (e.g. B is separable), one can use relative version of the invariants that we are interested in, which, in principle, are easier to compute. In the main part of my talk I will discuss the relationship between relative Hochschild and cyclic (co)homology and Hopf-cyclic (co)homology with coefficients.

DRAGOS STEFAN (University of Bucharest, Romania)


19 March 2007

EQUIVARIANT CHERN CHARACTER FOR TOTALLY DISCONNECTED GROUPS

We explain the construction of a bivariant Chern character from the equivariant KK-theory for a totally disconnected group to bivariant equivariant cohomology in the sense of Baum and Schneider. As a consequence, we obtain that the complexified left hand side of the Baum-Connes assembly map is given by Bredon homology. In the case of discrete groups, it yields the inverse to the Chern character for equivariant K-homology obtained by Lueck. A main ingredient in our approach is equivariant cyclic homology.

CHRISTIAN VOIGT (Universitaet Muenster, Germany)


26 March 2007

DUALITY IN EQUIVARIANT CYCLIC HOMOLOGY AND HOPF-CYCLIC COHOMOLOGY

In this talk, we explain how equivariant cyclic homology can be generalized to the setting of quantum groups. A basic feature of the resulting theory is an analogue of the Baaj-Skandalis duality isomorphism in KK-theory. To formulate and prove this duality, modular pairs in the sense of Connes and Moscovici play a crucial role. At the same time, this provides a link to Hopf-cyclic cohomology. Time permitting, we will indicate further possible connections.

CHRISTIAN VOIGT (Universitaet Muenster, Germany)


2 April 2007

SPECTRAL GEOMETRY OF QUANTUM LENS SPACES

I will focus on the case study of quantum lens spaces obtained from both the quantum SU(2) group and the Matsumoto twisted (isospectral) deformation of the 3-sphere. These noncommutative deformations will be compared with the classical spin geometry of the 3-sphere. Time permitting, I shall discuss other noncommutaive quotient spaces.

ANDRZEJ SITARZ (Uniwersytet Jagiellonski, Poland)


16 April 2007

NONCOMMUTATIVE SCHEMES AND THEIR GEOMETRIC REALIZATION

I will introduce the notions and basic properties of noncommutative schemes and more general locally affine noncommutative spaces, together with some motivating examples of different nature which triggered these notions and still continue to stimulate developments of noncommutative algebraic geometry. I will discuss underlying topological spaces and related geometric realizations of noncommutative schemes. These geometric realizations and surrounding them facts of noncommutative local algebra give new insight (and applications) to representation theory and to local study of morphisms of noncommutative schemes. They also produce natural tools for the study of K-groups of noncommutative schemes and more general noncommutative spaces. A commutative consequence of the geometric realizations is the reconstruction of commutative schemes from their categories of quasi-coherent sheaves. If time permits, I will shortly describe the triangulated version of this picture and/or mention some of its more exotic (non-additive) variants.

ALEXANDER ROSENBERG (IHES, France / Kansas State University, USA)


23 April 2007 (room 403)

RIGIDITY PHENOMENA FOR VON NEUMANN ALGEBRAS AND GROUP ACTIONS

Group actions on probability spaces give rise to von Neumann algebras. Recently, Sorin Popa has introduced very powerful techniques allowing in certain cases to recover the group and the action from the associated von Neumann algebra. I discuss several applications of these techniques: Popa's orbit equivalence superrigidity results, joint work with Popa on the outer automorphism groups of type II_1 factors, and my recent work on the existence of type II_1 factors without non-trivial finite index subfactors.

STEFAAN VAES (Katholieke Universiteit Leuven, Belgium)


7 May 2007

THE BAUM-CONNES CONJECTURE FOR QUANTUM GROUPS

I will present an approach to the Baum-Connes conjecture that involves localisation of categories. After briefly reviewing the classical case of groups, where we get the known Baum-Connes assembly map, I will discuss the case of coactions of compact groups and discuss the ingredients that seem relevant to construct an analogue of the Baum-Connes assembly map for general discrete quantum groups.

RALF MEYER (Universitaet Goettingen, Germany)


14 May 2007

UNBOUNDED OPERATORS AFFILIATED TO C*-ALGEBRAS

In many situations, we deal with unbounded operators that are in a sense related to operator algebras. This is the case in quantum mechanics, where (unbounded) Hamiltonian is related to the algebra of observables, and in the theory of locally compact (non-compact) quantum groups, where matrix elements of finite-dimensional representations are related to the C*-algebra of "functions on the group". Similarly, infinitesimal generators of a Lie group are related to the convolution C*-algebra of the group. In all these cases the unboundedness is the only feature that prevents us from including the operators in the C*-algebra. Instead, we say that the operators are affiliated with the algebra. The aim of the talk is to define the affiliation relation in the C*-algebraic context and show a number of applications and properties. (Based on the following articles: S.Baaj and P.Julg, S.L.Woronowicz, K.Napiorkowski and S.L.Woronowicz, S.L.Woronowicz.)

STANISLAW L. WORONOWICZ (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)


21 May 2007

GEOMETRIC STRUCTURE IN THE REPRESENTATION THEORY OF P-ADIC GROUPS

Let G be a reductive p-adic group. Examples are GL(n, F), SL(n, F), etc, where n can be any positive integer and F can be any finite extension of the p-adic numbers Q_p. The smooth dual of G is the set of equivalence classes of irreducible smooth representations of G. The relevant algebra for the smooth dual is the Hecke algebra of G. This is the convolution algebra of all locally constant compactly supported complex-valued functions f : G --->C. The Hecke algebra of G is a dense, but not holomorphically closed, sub-algebra of the reduced C*-algebra of G. The smooth dual contains the reduced unitary dual. This talk states a conjecture (due to A.-M.Aubert, P.Baum, and R.Plymen) which asserts that the smooth dual is a countable disjoint union of complex affine algebraic varieties. These varieties are explicitly identified. The conjecture is based on results of J.Bernstein, and if correct continues Bernstein's work. The inter-action with the reduced unitary dual (i.e. the support of the Plancherel measure) will be explained.

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


28 May 2007

CYCLIC HOMOLOGY WITH COEFFICIENTS THROUGH CORINGS

A new approach to the coefficients of cyclic homology will be presented using corings. In the case of the Sweedler coring, we obtain cyclic homology with coefficients of a ring extension with no assumption on commutativity of the base ring. The latter assumption was essential in the approach of Kaledin using the language of cocartesian objects in symmetric monoidal categories. It will be shown that in our formalism this restrictive context can be avoided, and cyclic homology with coefficients can be constructed in much greater generality.

TOMASZ MASZCZYK (Instytut Matematyki UW / Instytut Matematyczny PAN)


29 May 2007 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5810, 12:00 Tuesday.)

NONCOMMUTATIVE JOIN CONSTRUCTION

The aim of this talk is to show how to carry out the join construction of compact quantum groups avoiding braiding and replacing the unit interval by an arbitrary unital C*-algebra (noncommutative compact Hausdorff space). This is done in terms of equivariantly projective Hopf-Galois extensions of C*-algebras. The completion of the extended algebra is a natural candidate for a non-crossed product example of a principal extension of C*-algebras in the sense of Ellwood (non-trivial noncommutative principal bundle). The main point is a general and explicit formula for a strong connection, which puts us directly into the framework of the index pairing between K-theory and K-homology. (Based on a joint work with L.Dabrowski and T.Hadfield.)

PIOTR M. HAJAC (Instytut Matematyczny, Polska Akademia Nauk / Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)


4 June 2007

FIBRE-PRODUCT APPROACH TO THE INDEX PAIRING FOR HOPF FIBRATIONS

On the C*-algebraic level, the generic Podles quantum spheres are obtained by gluing two quantum discs along their boundaries (fibre product of Toeplitz algebras over the algebra of continuous functions on the circle). Applying the functional calculus, we will show that the noncommutative line bundles associated to the Hopf fibration of quantum SU(2) over a generic Podles sphere can be described as fibre products of Toeplitz algebras now viewed as free modules (trivial line bundles). Then, using explicit projections from the Bass construction, we easily determine the Chern numbers from the index pairing. Thus we obtain an alternative to the earlier index computation that relied on the noncommutative index formula. Finally, we argue that the entire construction still makes sense if one quantum disc is "shrinked to a point". On the technical level, this means that one Teoplitz algebra is replaced by the field of complex numbers and one of the mappings in the pull-back diagram is not surjective. This construction yields a description of the Hopf fibration of quantum SU(2) over the standard Podles sphere.

ELMAR WAGNER (Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski)