NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

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


5 October 2009

TOEPLITZ QUANTUM PROJECTIVE SPACES

We define the C*-algebra of a quantum complex projective space TP(n) as a multirestricted fiber product build from (n+1)-copies of the n-th tensor power of the Toeplitz algebra (Toeplitz cubes). Replacing the Toeplitz algebra by the algebra of continuous functions on a disc, one obtains the algebra of continuous functions on CP(n). Using Birkhoff's theorem on distributive lattices, we show that the lattice generated by the ideals defining this fibre product is free. This means that the fiber product structure is "maximally non-trivial" or, in geometric terms, that all possible intersections obtained from pieces of Toeplitz cubes covering this quantum projective space are non-empty. This is a property inherited from the affine covering of a projective space. All this is used as an example to illustrate the classification of finite closed coverings of compact quantum spaces by finitely supported flabby sheaves of algebras over the universal partition space (the infinite projective space over Z/2 equipped with the Alexandrov topology). Based on joint work with Atabey Kaygun and Bartosz Zieliński.

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)


12 October 2009

MORPHISMS BETWEEN FLABBY SHEAVES OF ALGEBRAS

We introduce a new type of morphisms between finitely-supported flabby sheaves of algebras over the universal partition space (infinite Z/2-projective space with the Alexandrov topology). These morphisms are obtained by taking a certain quotient of the usual class of morphisms enlarged by the actions of a specific family of endofunctors. Thus constructed morphisms yield a category of flabby sheaves that we prove to be equivalent to the category of (unordered) finite coverings of algebras. Based on joint work with Piotr M. Hajac and Atabey Kaygun.

BARTOSZ ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


19 October 2009

DEFORMATIONS OF MONOIDAL FUNCTORS

There is well-established theory of formal deformations of associative algebras in terms of the strong homotopy Gerstenhaber algebra structure on the Hochschild cochain complex of an algebra with coefficients in the algebra itself. It is easy to see that associative algebras are always images of monoidal units under some monoidal functors. We show how to lift the strong homotopy Gerstenhaber algebra structure from the case of associative algebras to the level of monoidal functors.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


26 October 2009

WHEN IS A QUANTUM SPACE NOT A QUANTUM GROUP?

Given a compact topological space one might pose the question whether it can be endowed with the structure of a topological group. In some cases it is simple to show that this cannot be done. Other cases require more effort. We address this question for objects of "noncommutative topology". We will show that some well-known quantum spaces like quantum tori and some quantum spheres cannot be given the structure of a compact quantum group. We will use completely different tools than those employed in classical topology or for polynomial algebras. In particular, we will be interested in the existence of characters and traces and nuclearity of C*-algebras describing the considered quantum spaces. At the end, we will mention some partial results on the question whether the quantum disk can be a compact quantum group.

PIOTR M. SOŁTAN (Uniwersytet Warszawski / IMPAN)


2 November 2009

THE CHERN-GALOIS CHARACTER AND EHRESMANN CYCLIC HOMOLOGY GROUPS

Principal bundles can be viewed as functors associating vector bundles to group representations. Combining such a functor with the Chern-Weil formalism allows one to compute invariants of vector bundles. A noncommutative-geometric generalization of this construction is a functor called the Chern-Galois character. It transforms quantum-group representations into cyclic homology classes. On the other hand, principal bundles give rise to the Ehresmann groupoids beautifully codifying their structure. A noncommutative version of this construction is a quantum groupoid devised by Peter Schauenburg. The talk will be focused on showing that the Chern-Galois character factorizes through cyclic homology groups intrinsically defined by the structure of the Ehresmann-Schauenburg quantum groupoid. This factorization gives a hope for finer invariants and shows unexpected links between the classical Chern character and Ehresmann groupoid. (Joint work with Gabriella Böhm.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)


9 November 2009 (Banach Center meeting on Categories, Hopf algebras, and noncommutative Galois theory.)

KLEISLI HOPF ALGEBRAS

Starting with a monoidal monad M on a braided monoidal category A, we consider the associated Kleisli category. This Kleisli category turns out to be again a braided monoidal category, and Hopf algebras in this new category will be termed Kleisli Hopf algebras. The purpose of this talk is to show that multiplier Hopf algebras and infinite Drinfel'd doubles arise as natural examples of our theory under a correct choice of M and A. This is joint work (in progress) with Kris Janssen.

JOOST VERCRUYSSE (Vrije Universiteit Brussel, Belgium)


10 November 2009 (Exceptional time: Tuesday. Banach Center meeting on Categories, Hopf algebras, and noncommutative Galois theory.)

GALOIS THEORY IN BICATEGORIES

We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal- Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application, we construct a new type of comatrix corings based on (dual) quasi bialgebras. (Joint work with Jose Gomez- Torrecillas.)

JOOST VERCRUYSSE (Vrije Universiteit Brussel, Belgium)


23 November 2009

PARTIAL (CO)ACTIONS OF HOPF ALGEBRAS AND PARTIAL HOPF-GALOIS THEORY

We introduce partial (co)actions of a Hopf algebra H on an algebra. To this end, we first introduce the notion of a lax coring that generalizes Wisbauer's notion of a weak coring. We will also discuss the dual notion of a lax ring, and explain several duality results. Then we will develop Galois theory for partial H-comodule algebras. Finally, a connection to the (weak) entwining structures will be clarified. (Joint work with S.Caenepeel.)

KRIS JANSSEN (Vrije Universiteit Brussel, Belgium)


26 November 2009 (Joint Noncommutative Geometry and Operator Algebras and Quantum Groups Seminar. Exceptional place and time: KMMF UW, ul. Hoża 74, 5th floor, 13:15 Thursday.)

GROUP CORINGS

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a structure theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra. (Joint work with S.Caenepeel and S.H.Wang.)

KRIS JANSSEN (Vrije Universiteit Brussel, Belgium)


30 November 2009 (Banach Center meeting on A Categorical Approach to Hopf algebras and their Cyclic Theories.)

A CATEGORICAL APPROACH TO CYCLIC DUALITY

We construct a large class of para-(co)cyclic objects from 2-functors of a canonically chosen domain. Their cyclic duality is shown to be governed by a functor between 2-functor categories. Examples of concrete realizations of our construction are provided by various para-(co)cyclic modules arising from Hopf-cyclic theory. (This is joint work with Dragoș Ștefan.)

GABRIELLA BÖHM (Hungarian Academy of Sciences, Budapest)


1 December 2009 (Exceptional time: Tuesday. Banach Center meeting on A Categorical Approach to Hopf algebras and their Cyclic Theories.

WEAK HOPF MONADS

A bialgebra over a field can be characterized as an algebra whose module category is monoidal with a strict monoidal forgetful functor to the category of vector spaces. By analogy, in the literature, a bimonad is defined as a monad on a monoidal category whose Eilenberg-Moore category of algebras is monoidal with a strict monoidal forgetful functor to the base category. More generally, we study weak bimonads, i.e., monads on monoidal categories, with a monoidal Eilenberg-Moore category, but requiring the forgetful functor to possess only compatible (non-strict) monoidal and op-monoidal structures. While for a bimonad the monoidal structure of the Eilenberg-Moore category is lifted from the base category, in the weak case it is obtained as a canonical retract. A weak bimonad on a Cauchy complete monoidal category is shown to be equivalent to a bimonad on another appropriately chosen monoidal category. A weak bimonad is said to be a weak Hopf monad provided that a canonical (`Galois type') natural transformation is an isomorphism. Examples of weak bimonads and weak Hopf monads are provided by weak bialgebras and weak Hopf algebras, respectively, in braided monoidal categories. (This is joint work in progress with Stephen Lack and Ross Street.)

GABRIELLA BÖHM (Hungarian Academy of Sciences, Budapest)


7 December 2009

QUANTUM SETS AND QUANTUM GALOIS GROUP

I will start from describing a strong monoidal faithful and full embedding of the Cartesian monoidal category of sets into 1-cells of some 2-category. These 1-cells are op-monoidal functors (admitting the right adjoint) between some monoidal categories ("quantum vector spaces over quantum fields"). The image of this embedding will be regarded as picking classical sets among their quantum counterparts. Next, I will show how to endow the (convolution) representations of "quantum fields" (some associative unital algebras) with a structure of an enriched category whose morphisms form "quantum sets" as above. This category should be understood as "quantum category of quantum fields". It generalizes its classical counterpart, in particular the classical Galois group.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


14 December 2009

SPECTRAL TRIPLES WITH TORSION

In classical differential geometry and general relativity the torsion tensor is usually assumed to vanish. In the framework of noncommutative geometry there is, however, no good replacement for such a condition. Therefore, spectral triples for Dirac operators with torsion are a priori admissible. I shall discuss the general construction of Dirac operators and restrictions on torsion arising from Connes' axioms of spectral triples, and illustrate all this by commutative and noncommutative examples. Finally, I shall present the main result that is the construction of spectral action for Dirac operators with torsion.

ANDRZEJ SITARZ (Uniwersytet Jagielloński, Kraków)


18 January 2010

RESIDUE FORMULAS AND INDEX PAIRINGS FOR THE STANDARD PODLEŚ SPHERE

In the general framework of noncommutative geometry, residue formulas are used to associate cyclic cocycles to (regular) spectral triples and compute index pairings. Applying these ideas to the 0-summable spectral triple on the standard Podleś sphere, one faces two problems. First, the spectral triple fails the regularity condition, which is a prerequisite for the development of a pseudo-differential calculus and the definition of "local" index formulas. Next, the Hochschild and cyclic cohomologies are in some sense degenerated - one needs twisted versions of these cohomology theories to obtain good correspondence to the classical case. To deal with these problems, we present the definition of a twisted Chern character from equivariant K_0-theory into twisted cyclic homology, give residue formulas for some distinguished (twisted) cocycles on the standard Podleś sphere, and then compute the index pairing. (Joint work with Ulrich Krähmer.)

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


15 February 2010

THREE CONJECTURES IN THE REPRESENTATION THEORY OF REDUCTIVE p-ADIC GROUPS (PART I)

These three talks will consider three conjectures and the interactions among them. The three are: #1. Local Langlands, #2. Baum-Connes, #3. Aubert-Baum-Plymen. The conjectures will be stated and then the point of view will be developed that Aubert-Baum-Plymen provides a connection between Local Langlands and Baum-Connes. The three talks are intended to be accessible to a general mathematical audience. Thus an introduction to the representation theory of reductive p-adic groups will be included in the talks.

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


22 February 2010

THREE CONJECTURES IN THE REPRESENTATION THEORY OF REDUCTIVE p-ADIC GROUPS (PART II)

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


1 March 2010

THREE CONJECTURES IN THE REPRESENTATION THEORY OF REDUCTIVE p-ADIC GROUPS (PART III)

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


8 March 2010

C*-CORRESPONDANCES, THEIR C*-ALGEBRAS AND APPLICATIONS (PART I)

We discuss constructions of C*-algebras related to C*-correspondences (or Hilbert bimodules) and, more generally, product systems of Hilbert bimodules. As an application of these methods, we show how mirror quantum spheres correspond to: (i) pullbacks of C*-correspondences, (ii) labelled graphs. Then we indicate how product systems of Hilbert bimodules may be applied to the regular C*-algebras of integral domains, as defined by Cuntz. The latter are, in turn, related to the Bost-Connes systems and the Toeplitz algebras of certain arithmetic semigroups.

WOJCIECH SZYMAŃSKI (Syddansk Universitet, Odense, Denmark)


9 March 2010 (Exceptional time: Tuesday.)

C*-CORRESPONDANCES, THEIR C*-ALGEBRAS AND APPLICATIONS (PART II)

WOJCIECH SZYMAŃSKI (Syddansk Universitet, Odense, Denmark)


15 March 2010

EXAMPLES OF NON-COMPACT QUANTUM GROUP ACTIONS

We will describe locally compact group actions on C*-algebras in a language suitable for a noncommutative generalization. Then we will give two examples of homogeneous spaces for non-regular quantum groups. We will analyze these examples and show that the resulting actions of our quantum groups on their homogeneous spaces are continuous.

PIOTR M. SOŁTAN (Uniwersytet Warszawski / IMPAN)


29 March 2010

THE RIEFFEL DEFORMATION OF HOMOGENEOUS SPACES

Let H be a closed subgroup of a locally compact group G, and let X be the quotient space of cosets. Also, let L be an abelian closed subgroup of H and f be a 2-cocycle on the dual group of L. Using these data, we can define an action of the quantum group G(f) (the Rieffel deformation of G) on the Rieffel deformation of X. On the other hand, we can perform the Rieffel deformation of the subgroup H to obtain the quantum subgroup H(f) of G(f). This in turn, by the results of Vaes, leads to the C*-algebraic quotient G(f)/H(f). The main aim of this talk is to show that the two construction described above give isomorphic quantum homogeneous spaces. Finally, we also consider the case where L is a subgroup G but not of H. Then we cannot construct the quantum subgroup H(f), and X(f) is not a quotient space. However, we will show that the action of the quantum group G(f) on the noncommutative space X(f) is minimal.

PAWEŁ KASPRZAK (Uniwersytet Warszawski / Syddansk Universitet, Odense, Denmark)


12 April 2010

HOPF-CYCLIC COHOMOLOGY IN BRAIDED MONOIDAL CATEGORIES

We extend the formalism of Hopf-cyclic cohomology to the context of braided categories. We introduce the notion of a stable anti-Yetter-Drinfeld module for a Hopf algebra in a braided monoidal abelian category. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution. Thus we obtain a braided generalization of the original construction of Connes and Moscovici. When the braiding is symmetric, the full formalism of Hopf-cyclic cohomology with coefficients can be extended to our categorical setting. (This is joint work with Masoud Khalkhali.)

ARASH POURKIA (University of Western Ontario, London, Canada)


19 April 2010

NONCOMMUTATIVE GEOMETRY AND HARMONIC ANALYSIS ON THE HALF-PLANE

What is the most natural non-commutative structure over a given manifold? For ordinary phase spaces, the answer is known: the Moyal algebra. It is an object of much interest to physicists these days. Together with the standard Dirac operator, it defines a non-compact non-commutative manifold in a suitably narrow sense. A case can be made for generalizing this to orbits of coadjoint actions, which are natural sites for the meeting of the Kirillov and Connes paradigms. We show how to define thus generalized Moyal algebra and putative spectral triples on the Poincare half-plane. We also discuss nice surprises (ranging from one-variable real calculus to non-commutative harmonic analysis) that crop up along the way.

JOSE M. GRACIA-BONDIA (Universidad de Zaragoza, Spain)


26 April 2010

SOURCES OF STABLE ANTI-YETTER DRINFELD MODULES

Stable anti-Yetter-Drinfeld modules are the coefficients of Hopf-cyclic cohomology. In this talk, we review the known sources of stable anti-Yetter-Drinfeld modules and reveal some new places where these modules naturally appear. The first time these coefficients were used was in the work of Connes and Moscovici on the local index formula. Another application was in the work of Jara and Stefan on relative cyclic cohomology and Hopf-Galois extensions. We show that there are interesting stable anti-Yetter-Drinfeld modules coming from decompositions of Lie algebras and from Hopf-Galois coextensions. We end the talk by generalizing the latter source of interesting examples. Part of this talk is based on joint work with my students R.G.Shoushtari and S.Sutlu.

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


27 April 2010 (Exceptional time: Tuesday.)

HOPF-CYCLIC COHOMOLOGY OF TYPE III HOPF ALGEBRAS

For any positive integer n, we introduce a Hopf algebra K(n) that is responsible for the geometric part of the cyclic cohomology of the crossed product of the the algebra of compactly supported functions on a manifold of dimension n with the group of its diffeomorphisms. We upgrade our computing tools for the Hopf-cyclic cohomology of Hopf algebras of type II, and completely determine the Hopf-cyclic cohomology of K(n). This is joint work with Henri Moscovici.

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


10 May 2010

EXPLICIT FORMULAS FOR THE HAAR FUNCTIONAL ON SUq(N)

The work presented here was conceived by trying to extend the work of Dabrowski, Landi, Sitarz, van Suijlekom, and Várilly from a pair of papers written about the Dirac operator and Local Index Theorem for SUq(2). The first stumbling block came with computing the Haar functional and producing an orthonormal basis for a Hilbert space on which the proposed Dirac operator should act. Luckily, with the assurance of Woronowicz's theorem that there exists a unique normalizable functional which is bi-invariant one can proceed. From the assurance that such a functional exists and knowing the answer for N=2 one can attempt to compute for N>2. The present work takes the work done on matrix corepresentations for N=2 and extends it using nearly identical methods. The matrix corepresentations are now classified by a single number rather than an n-tuple. The result is an explicit computation of the Haar functional for SUq(N) with the "same" result as for SUq(2). Furthermore, an amusing conjecture about particle spin in higher dimensions arises.

CLARK ALEXANDER (The Institute of Mathematical Sciences, Chennai, India)


24 May 2010

THE WEYL CHARACTER FORMULA AND KK-THEORY

Weyl's formula describes the characters of the irreducible representations of compact Lie groups. It has a beautiful relationship with K-theory and index theory, as was pointed out by Atiyah and Bott a long time ago. I shall revisit the subject, partly in order to give a new introductory account of Kasparov's KK-theory, and partly to indicate a newly emerging connection between the Baum-Connes conjecture and geometric representation theory.

NIGEL HIGSON (Pennsylvania State University, State College, USA)


31 May 2010

EXAMPLES OF PAIRINGS IN HOPF-CYCLIC COHOMOLOGY

I will give some explicit examples of pairings between the Hopf-cyclic cohomology of algebras and coalgebras in case when they both arise from geometric constructions. My examples will include the universal enveloping algebras of vector fields and group algebras acting on functions on manifolds, etc. I will also consider a bar-resolution of the de Rham algebra of a manifold coacting on the de Rham algebra of a principal bundle over this manifold with the coaction given by a twisting cochain associated with the bundle. In all these cases, I shall try to express the pairing in question in more classical (geometric) terms.

GEORGY SHARYGIN (ITEP, Moscow, Russia)