NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Śniadeckich 8, room 322, Mondays

1999/2001 2002/2003 2003/2004 2004/2005 2005/2006 2006/2007 2007/2008 2008/2009 2009/2010 2010/2011 2011/2012

15 October 2012, 10:15-12:00

THE K-THEORY OF HEEGAARD QUANTUM LENS SPACES

Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex line bundle. The main result is the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the odd-to-even connecting homomorphism to prove the stable non-triviality of the bundles. (Joint work with Adam Rennie and Bartosz Zieliński.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)

15 October 2012, 14:15-16:00

THE K-THEORY OF THE TRIPLE-TOEPLITZ DEFORMATION OF THE COMPLEX AND REAL PROJECTIVE PLANES

Starting from the affine covering of a projective space, we construct a noncommutative deformation of both complex and real projective spaces in terms of Pedersen's multi-pullback of C*-algebras. In dimension 2, we obtain triple-pullback C*-algebras, and for such C*-algebras we devise a certain straightforward method to compute the K-groups. This talk is focused on the real case, where one also needs an explicit formula for the even-to-odd connecting homomorphism in the Mayer-Vietoris six-term exact sequence. We prove from scratch such a general formula, and then apply it to determine the K-theory of the aforementioned noncommutative deformation of the real projective plane. (Joint work with Paul F. Baum and Piotr M. Hajac.)

JAN RUDNIK (IMPAN)

22 October 2012, 10:15-12:00

MODULAR FREDHOLM MODULES AND TWISTED COCYCLES

Connes and Cuntz proved that cyclic cocycles have a geometric interpretation as characters of finitely-summable Fredholm modules. An analogous result can be obtained for twisted cyclic cohomology using the Chern character of modular Fredholm modules. General case will be illustrated by examples of modular Fredholm modules arising from Podleś spheres and from SUq(2). The talk is based on joint work with Adam Rennie and Makoto Yamashita.

ANDRZEJ SITARZ (Uniwersytet Jagielloński, Kraków, Poland)

22 October 2012, 14:15-16:00

THE MODULAR CLASS OF QUANTUM PERMUTATIONS

We introduce a new notion of a Radon-Nikodym differentiable structure on a finite-dimensional algebra. This allows us to speak about a quantum fundamental class on a finite quantum space. For any coaction of a Hopf algebra with bijective antipode on a finite-dimensional algebra with a fundamental class, we ask whether it preserves the fundamental class, as it is the case in classical geometry. If the answer is positive, there exists a canonical cohomology class that is an obstruction to the existence of an invariant Frobenius structure. We call this cohomology class the modular class. For the universal coaction, it provides an invariant of Frobenius algebras. We show that the universal coaction of a Hopf algebra with bijective antipode on the algebra of functions on a finite set preserves the fundamental class and that, for a finite set containing at least two elements, the modular class is non-trivial.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)

29 October 2012, 10:15-12:00

EMBEDDABLE QUANTUM HOMOGENEOUS SPACES

In this talk various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups will be discussed. On the von Neumann algebra level, we find an interesting duality for such objects which may be treated as a quantum version of the Takesaki-Tatsuuma duality. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is discussed in detail, as it seems to be a natural candidate for a quantum analogue of the classical homogeneous spaces. Among various examples, we single out the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup and quotients by compact quantum subgroups. Based on joint work with Piotr M. Sołtan.

PAWEŁ Ł. KASPRZAK (Uniwersytet Warszawski / IMPAN)

29 October 2012, 14:15-16:00

SPECTRAL TRIPLES ON CROSSED PRODUCTS BY EQUICONTINUOUS ACTIONS

A Kasparov-product inspired method of constructing spectral triples on crossed products by actions of discrete groups is discussed. If the original spectral triple is Lipschitz regular (in the sense of Rieffel), a sufficient condition for the method to work for actions of Z (independently introduced and applied by Jean Belissard, Mathilde Marcolli and Kamran Reihani) is identified with the topological equicontinuity of the action. We discuss certain examples and further related problems. (Joint work with Andrew Hawkins, Stuart White and Joachim Zacharias.)

ADAM SKALSKI (IMPAN / Uniwersytet Warszawski)

5 November 2012, 10:15-12:00

SPECTRAL PROPERTIES OF THE STANDARD PODLEŚ QUANTUM SPHERE

In 2003, Dąbrowski and Sitarz provided an interesting example of a spectral triple on the algebra of the standard Podleś sphere with a Dirac operator depending on the deformation parameter q. The analysis of spectral zeta functions associated with this Dirac operator revealed a peculiar dimension spectrum containing second order complex poles. This fact combined with the 0-summability and non-regularity makes the standard Podleś sphere a rather degenerate noncommutative space. Nevertheless, applications to physics via the spectral action and heat kernel methods are possible. The talk is based on joint work with Bruno Iochum and Andrzej Sitarz.

MICHAŁ ECKSTEIN (Uniwersytet Jagielloński, Kraków, Poland)

5 November 2012, 14:15-16:00

BUNDLES OVER TRIPLE-PULLBACK QUANTUM PROJECTIVE PLANES

Complex and real projective spaces are often defined as the quotients of spheres by appropriate free actions. The affine coverings locally trivialise these non-trivial principal bundles. Taking compact refinements of these coverings, for both complex and real projective planes, we construct noncommutative deformations that preserve the piecewise trivial bundle structure. In the real case, the projective plane is obtained by first gluing two squares to the Möbius strip, and subsequently attaching the third square to the boundary of the Möbius strip. The 2-sphere can then be viewed as the surface of a cube, or as a gluing of the three pairs of squares. The antipodal Z/2Z-action trivialises over each square yielding a piecewise trivial principal bundle. Using the Toeplitz algebra, we deform the squares while retaining the gluing recipies translated into Pedersen's multi-pullbacks of C*-algebras. Taking advantage of the stable non-triviality of the associated quantum tautological line bundle (proved with the help of K-theory methods by Baum, Hajac and Rudnik), we conclude the non-cleftness of the thus obtained principal comodule algebra. The complex case is treated much in the same way yielding a Heegaard-type quantum 5-sphere over the triple-Toeplitz deformation of the complex projective plane. (Based on joint work with Jan Rudnik and Bartosz Zieliński.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)

19 November 2012, 14:15-16:00

DISTANCE, TIME AND SPECTRAL TRIPLES FOR LORENTZIAN NONCOMMUTATIVE GEOMETRY

We will present some attempts to generalize Connes' theory of noncommutative geometry in order to deal with Lorentzian signature spacetimes. We will show how the construction of the Riemannian distance can be adapted to obtain a path independent formula for the distance on Lorentzian manifolds. Finally, it will be explained how to construct Lorentzian spectral triples and how such a structure can admit a global time element.

NICOLAS FRANCO (Copernicus Center for Interdisciplinary Studies, Kraków, Poland)

3 December 2012, 10:15-12:00

THE TAUTOLOGICAL LINE BUNDLE OVER THE TRIPLE-PULLBACK QUANTUM COMPLEX PROJECTIVE PLANE

We prove that the module of the tautological line bundle over the triple- pullback quantum complex projective plane is not stably free. Our tools are a strong connection on the Heegaard quantum 5-sphere and the Milnor odd-to-even connecting homomorphism in K-theory. The former yields an explicit idempotent representing the module, and the latter leads to an equally explicit idempotent of a non-trivial generator of a certain K-group. Inserting these formulae to an iterated pullback diagram computing the K-groups of our quantum complex projective plane allows us to infer the result. On the way, we determine the aforementioned K-groups, and see that the tautological line bundle is partially responsible for the non-trivial part of the even group. On the other hand, we conclude that the U(1)-C*-algebra of the Heegaard quantum 5-sphere is not a crossed-product of the fixed-point subalgebra and the group of integers. This means that the compact quantum principal U(1)-bundle of the Heegaard 5-sphere over the quantum complex projective plane is not trivial. (Based on joint work with Jan Rudnik.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)

17 December 2012, 10:15-12:00

EXAMPLES OF MODULAR FREDHOLM MODULES

I will present examples of modular Fredholm modules for SUq(2) and the family of Podleś spheres. This will include the "standard" modular Fredholm modules (based on irreducible representations) as well as modular Fredholm modules arising from spectral triples. The latter construction shall be used to discuss how to find generators of the C*-algebra of the Heegaard and mirror quantum spheres that would be suitable to describe their geometry.

ANDRZEJ SITARZ (Uniwersytet Jagielloński, Kraków, Poland)

17 December 2012, 14:15-16:00

CLASSIFICATION OF QUANTUM HOMOGENEOUS SPACES

We study actions of compact quantum groups on operator algebras from the categorical point of view. The ergodic actions admit a particularly nice classification in terms of the category of equivariant Hilbert modules. For the case of quantum SU(2), we obtain a simple description of equivariant homomorphisms and K-groups based on the categorical structure of such modules. Based on joint work with Kenny De Commer.

MAKOTO YAMASHITA (Ochanomizu University, Tokyo, Japan)

14 January 2013, 10:15-12:00

UNSOLVED PROBLEM FOR QUANTUM "ax+b"-GROUPS

We analyse quantum "ax+b"-groups in the setting of C*-algebras. In this case, the deformation parameter q is of modulus 1, and we write its square as exp(-iħ). Our considerations are restricted to small positive values of ħ. Besides a, its inverse and b one has to use some additional elements to generate the algebra of "functions" on the quantum group. We introduce an integer parameter called the size of the group measuring how much the algebra is enlarged by the additional generators. We shall show that quantum "ax+b"-groups of minimal size appear only if the deformation parameter ħ=π/(2k+3), where k=0,1,2,.... The unsolved problem is the lack of a similar result for the quantum "ax+b"-groups of finite size.

STANISŁAW L. WORONOWICZ (Uniwersytet Warszawski)

14 January 2013, 14:15-16:00

QUANTUM GROUPS, L^p-SPACES AND FOURIER THEORY

We introduce Fourier transforms on noncommutative L^p-spaces associated with a quantum group. We show that it is imperative to use the techniques of L^p-spaces constructed out of type III von Neumann algebras even if we are dealing with a semi-finite case. We also discuss spherical analogues of the Fourier transform and describe it explicitly in the case of "extended SUq(1,1)". Finally, commutator estimates in noncommutative L^p-spaces (obtained jointly with D. Potapov, S. Montgomery-Smith and F. Sukochev) will be presented time permitting.

MARTIJN CASPERS (Université de Franche-Comté, Besançon)

18 February 2013, 10:15-12:00

DEFORMATION QUANTIZATION OF HAMILTONIAN ACTIONS IN POISSON GEOMETRY

We recall the notion of a momentum map in Poisson geometry and describe the so-called Poisson reduction. The latter is a technique that allows one to reduce the dimension of a manifold in the presence of symmetries implemented by Poisson actions. Using the tools of deformation quantization and quantum groups, we define a quantum momentum map in such a way that it generates quantum actions.

CHIARA ESPOSITO (Universitat Autònoma de Barcelona)

18 February 2013, 14:15-16:00

C*-ALGEBRAS GENERATED BY q-NORMAL OPERATORS

S. L. Woronowicz's theory of generating C*-algebras by unbounded operators is applied to q-normal operators satisfying the defining relation of the quantum complex plane. A non-unital C*-algebra generated in this sense can be interpreted as the algebra of continuous functions vanishing at infinity on the quantum complex plane. It is a subalgebra of the crossed product of the algebra of the continuous functions vanishing at infinity on the half line and the group of integers. Its unitization is considered as the C*-algebra of continuous functions on a quantum 2-sphere. Differences between this construction and the construction of Podleś spheres will be highlighted. Finally, the K-theory and K-homology of the new quantum 2-sphere will be discussed.

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

25 February 2013, 10:15-12:00

ON CERTAIN LIMIT CONSTRUCTIONS FOR QUANTUM SYMMETRY GROUPS

To every C*-algebra equipped with an orthogonal filtration one can associate its quantum symmetry group. In this talk, I will recall the details of the relevant construction (developed jointly with Teo Banica), introduce some examples and then describe a certain natural behaviour of the corresponding quantum symmetry groups with respect to inductive limits of C*-algebras with orthogonal filtrations. The quantum symmetry group arising in the limit will be shown to be the projective limit of quantum symmetry groups constructed in consecutive steps. I will then present some applications of this result and also discuss the case, related to duals of the symmetric groups, where the theorem does not apply but the projective limit of the quantum symmetry groups can still be explicitly described. The work is partially inspired by the earlier joint work with Teo Banica, Jyotishman Bhowmick, Debashish Goswami, Piotr Sołtan and Jan Liszka-Dalecki. (Joint work with Piotr Sołtan.)

ADAM SKALSKI (IMPAN / Uniwersytet Warszawski)

4 March 2013, 10:15-12:00

COMMON GENERALIZATION OF CUNTZ-PIMSNER AND DOPLICHER-ROBERTS C*-ALGEBRAS

Cuntz-Pimsner algebras were introduced as a common generalization of crossed products and graph C*-algebras, while Doplicher-Roberts algebras arise as "simple duals" in abstract duality for non-commutative compact groups. We present a new machinery that generalizes and unifies these two constructions. The machinery is based on the language of C*-precategories (a categorial analogue of not necessarily unital C*-algebras) and their ideals. Our approach allows a "matrix construction" of the aforementioned algebras and sheds a new light on their structure.

BARTOSZ K. KWAŚNIEWSKI (Uniwersytet w Białymstoku)

11 March 2013, 10:15-12:00

ANTIPODES FOR HOPFISH ALGEBRAS

The notion of a Hopfish algebra was introduced by Tang, Weinstein and Zhu. From the viewpoint of non-commutative geometry, it is a very natural generalization of a Hopf algebra. In contrast with Hopf algebras, a Hopfish algebra has as structural maps bimodules rather than homomorphisms of algebras. Although the bialgebra conditions are easily transferred to this setting, the antipode condition is more problematic. The one imposed by Tang, Weinstein and Zhu is motivated and justified by Poisson geometry, but it seems to lack good properties. In this talk, we want to propose a different notion of antipode which has more structural content. This is work in progress, joint with J. Vercruysse.

KENNY DE COMMER (Université de Cergy-Pontoise)

11 March 2013, 14:15-16:00

K-CYCLES FOR TWISTED K-HOMOLOGY

Let X be a locally finite CW complex with a given twisting datum. Here "twisting datum" is a vector bundle of C* algebras on X in which each fiber is an elementary C* algebra. An elementary C* algebra is a C* algebra A such that there exists a Hilbert space H and an isomorphism of A to the C* algebra of all compact operators on H. This talk will introduce K-cycles for the K-homology of X twisted by the given twisting datum. The abelian group of these K-cycles is isomorphic to the Kasparov twisted K-homology of X. Thus a context for twisted index theory is obtained. Twisted K-cycles are very closely connected to the D-branes of string theory. The above is joint work with A. Carey and B.-L. Wang.

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

15 April 2013 10:15-12:00

CONTRACTIVE IDEMPOTENTS ON LOCALLY COMPACT QUANTUM GROUPS

A contractive idempotent on a locally compact group is a norm 1 measure on the group such that the measure is an idempotent under the convolution product. F. P. Greenleaf characterised such measures as twists by continuous characters of the Haar measures of compact subgroups. We shall discuss a generalisation of Greenleaf's result that gives an analogous characterisation for contractive idempotents on coamenable locally compact quantum groups. We shall also consider structures related to contractive idempotents. Namely, every contractive idempotent determines a convolution operator whose image is a ternary ring of operators (TRO) in the C*-algebra of "continuous functions vanishing at infinity" associated to the underlying quantum group. Conversely, certain TROs necessarily arise from contractive idempotents, leading to correspondence results under some conditions. The talk is based on joint work with Matthias Neufang, Adam Skalski and Nico Spronk.

PEKKA SALMI (Oulun yliopisto)

15 April 2013 14:15-16:00

NON-MINIMAL LAPLACE-TYPE OPERATORS ON NONCOMMUTATIVE TORI

There are several new candidates for Dirac and Laplace-type operators on noncommutative tori that appear to provide a truly curved noncommutative geometry. In some examples studied so far, the scalar curvature functional was shown not to vanish. However, it is not yet clear whether the proposed operators (which are direct generalizations of their classical counterparts) are minimal. For instance, they might include torsion-like terms and fail being minimal. To settle the issue, I put forward a condition based on heat-kernel asymptotics and discuss its application to two families of operators over noncommutative tori.

ANDRZEJ SITARZ (Uniwersytet Jagielloński / IMPAN)

22 April 2013 10:15-12:00

GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY

We introduce and study the fundamental quantum group of a graph of quantum groups and prove that, if the initial quantum groups are amenable, then the fundamental quantum group is K-amenable. This generalizes to the quantum setting the result of Julg and Valette about the K-amenability of a group acting on a tree with amenable stabilizers.

PIERRE FIMA (Université Paris Diderot-Paris 7)

22 April 2013 14:15-16:00

EXPLICIT PRESENTATION OF CHARACTERISTIC CLASSES OF FOLIATIONS IN HOPF-CYCLIC COHOMOLOGY

The talk is divided into two parts. First, we develop a new cohomology theory that we call equivariant Hopf-cyclic cohomology. We use it to detect a unique SAYD-twisted cyclic cocycle over a groupoid action algebra. Then we apply this cocycle to transform the characteristic classes of foliations into cyclic cocycles on the groupoid action algebra. In the second part, we develop a Hopf version of the truncated Weil algebra of the general linear group and a new characteristic map by which we transform the characteristic classes of foliations into b-B cocycles on the groupoid action algebra. At the end, we show that the results of the two methods coincide on the level of cyclic cohomology. The first part is joint work with Serkan Sutlu, and the second part is based on collaboration with Henri Moscovici.

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

13 May 2013 10:15-12:00

QUARTIC MATRIX MODEL AND 4-DIMENSIONAL NONCOMMUTATIVE QUANTUM FIELD THEORY

We study quartic matrix models with the action trace(EM²+(λ/4)M⁴), where M is a Hermitean NxN-matrix. The external matrix E encodes the dynamics, and λ>0 is a scalar coupling constant. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula that gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation, which must be solved case by case for a given E. These results imply that, if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function. As a main application, we prove that Euclidean φ⁴-quantum field theory on the 4-dimensional Moyal space is, in the limit θ→∞, exactly solvable and non-trivial. Using the theory of singular integral equations of Carleman type, we reduce the determination of the 2-point function to the solution of a non-linear integral equation. The existence of a solution is established via the Schauder fixed point theorem.

RAIMAR WULKENHAAR (Universität Münster)

20 May 2013, 10:15-12:00

EXTENDED QUOTIENT

Let G be a group acting on a set X. The quotient X/G is obtained by collapsing each orbit to a point. The extended quotient, denoted X//G, is obtained by replacing each orbit by the set of conjugacy classes of the isotropy group of any point in the orbit. The extended quotient of the second kind, denoted (X//G)₂, is obtained by replacing each orbit by the set of (equivalence classes of) irreducible representations of the isotropy group of any point in the orbit. This talk will explore some applications of extended quotients. The applications are to the equivariant Chern character and to representation theory of Lie groups and p-adic groups. In representation theory, the issue is how to "lift" the Baum-Connes conjecture from K-theory to representation theory.

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

20 May 2013, 14:15-16:00

C*-QUANTUM GROUPS WITH A PROJECTION

C*-quantum groups with a projection are the quantum-group analogue of semidirect products of groups. In parallel to Radford's theorem for Hopf algebras with a projection, we describe a C*-quantum group with a projection using a quantum group in the braided tensor category of coactions of a certain quasitriangular C*-quantum group. (Joint work in progress with Ralf Meyer and Stanisław Lech Woronowicz.)

SUTANU ROY (Universität Göttingen)

27 May 2013, 10:15-12:00

DEFORMATION QUANTIZATION OF INTEGRABLE SYSTEMS

For a Poisson-commutative subalgebra in the algebra of functions on a Poisson manifold, we want to find a deformation quantization of the Poisson manifold that preserves the commutativity of the subalgebra. We show that a sufficient condition for this deformation to exist is expressible in the terms of certain classes in the Poisson homology of the manifold. We also discuss the necessary existence conditions and formality properties of the DG Lie algebras related to this problem. Based on joint work with D. Talalaev (ITEP).

GEORGY SHARYGIN (ITEP, Moscow, Russia)

27 May 2013, 14:15-16:00

CHARACTERISTIC CLASSES OF COMBINATORIC BUNDLES

This talk reports on a long-term joint project with N. Mnev (POMI, St. Petersbourgh). We study the combinatoric spherical bundles, i.e. simplicial complexes with maps from other simplicial complexes into them that are locally trivial (in some sense) and have fibres PL-isomorphic to spheres. Given a combinatoric description of a spherical bundle, we apply a combinatorial method to describe the simplicial (co)chains representing Pontryagin classes. This method is based on suitable generalisations of Igusa's higher Reidemeister-Franz torsion, and the combinatorics of the twisting cochain of a bundle, combined with cyclic homology and other tools. In particular, with the help of the Gauss functor, our approach yields a construction of the Pontryagin classes of a combinatoric manifold.

GEORGY SHARYGIN (ITEP, Moscow, Russia)

3 June 2013, 10:15-12:00

CARTESIAN PRODUCTS IN NONCOMMUTATIVE GEOMETRY

Motivated by the fundamental role of the Cartesian product in commutative geometry, various approaches have been proposed to enable one to perform similar constructions with noncommutative algebras. Braided monoidal categories, distributive laws for monads of John Beck, factorization structures of Shahn Majid, to mention the most seminal ideas, generalize the classical Cartesian product. In this talk, we are going to sketch the panorama of technical problems related to these approaches. In particular, we will focus on the concept of a twist treated as a local braid, coherence theorems and their relation with the Yang-Baxter equation, the Drinfeld twist, the Martini product, and the problem of defining Cartesian products by their universal properties. The case of the Durdevic twist for Hopf-Galois extensions will be discussed in detail. Then we shall argue that, in the case of torsors, different twists for Hopf algebras have to be considered at the same time. Finally, we will propose a new categorical approach needed to unify these phenomena. As an application we shall present some quantum geometric constructions made possible in this approach, especially the quantum group law of Hopf algebras in terms of quantum points. We will also show how some constructions with free amalgamated products fit this framework.

TOMASZ MASZCZYK (Uniwersytet Warszawski)