NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

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


7 October 2008 (Joint Noncommutative Geometry and Algebraic Topology Seminar. Exceptional place and time: Instytut Matematyki UW, ul. Banacha 2, room 5870, 12:00 Tuesday.)

THE PETER-WEYL-GALOIS THEORY OF CARTAN COMPACT PRINCIPAL BUNDLES

We define a functor from the category of unital C*-algebras with compact quantum group actions to the category of comodule algebras by extending the notion of the algebra of regular functions (spanned by the matrix coefficients of the irreducible unitary corepresentations) from compact quantum groups to unital C*-algebras on which they act. We call it the Peter-Weyl functor. Combined with the Gelfand transform, it translates compact group actions on compact Hausdorff spaces into a general algebraic framework. On the other hand, the Galois condition for finite field extensions is also translated into this comodule-algebraic setting, and is the founding stone of noncommutative Hopf-Galois theory. The talk will be focused on showing the equivalence of the freeness of a classical compact group action on a compact Hausdorff space and the Galois condition for its Peter-Weyl comodule algebra. This result paralles the well-known equivalence of Galois coverings and discrete group principal bundles. (Based on joint work with P.F.Baum, U.Kraehmer, R.Matthes, and B.Zielinski.)

PIOTR M. HAJAC (IMPAN / Uniwersytet Warszawski)


13 October 2008

DISTRIBUTIVE LATTICES AND COHOMOLOGY

A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of modules over Dedekind rings. A generalized Chinese Remainder Theorem is derived as a consequence of the above resolution. The Gelfand-Naimark duality between finite closed coverings of compact Hausdorff spaces and the generalized Chinese Remainder Theorem is clarified.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


20 October 2008

QUANTUM GROUP AND SEMIGROUP ACTIONS ON M_2(C)

Using the universal description of quantum SO(3) groups I will give a complete classification of compact quantum group actions on the algebra of matrices M_2(C). Then I will extend this classification to a large class of much more general quantum semigroup actions (not necessarily compact).

PIOTR M. SOŁTAN (Uniwersytet Warszawski)


27 October 2008

KONTSEVICH GRAPH COMPLEX AND LEIBNIZ HOMOLOGY

Kontsevich has proven that the Lie homology of the Lie algebra of symplectic vector fields can be computed in terms of the homology of a graph complex. We prove that the Leibniz homology of this Lie algebra can be computed in terms of the homology of a variant of the graph complex endowed with an action of the symmetric groups. The resulting isomorphism is shown to be a Zinbiel-associative bialgebra isomorphism.

EMILY BURGUNDER (IMPAN, European Postdoctoral Institute)


3 November 2008

PARKING FUNCTIONS AND TRIDENDRIFORM BIALGEBRA STRUCTURES

A Parking function is a sequence of n positive integers majorated by a permutation of {1,...,n}. We consider a family of Hopf algebras that can be endowed with a finer structure, namely a shuffle algebra structure. We show that the Hopf algebra of Parking functions is of the same type. Moreover, as a tridendriform bialgebra, the latter is cofree and satisfies the generalised bialgebra structure theorem. We unravel the structure of its primitives: it is a variation of a Gerstenhaber-Voronov algebra. Then, we construct a family a q-tridendriform algebras and, under a cofreeness condition, we determine the structure of their primitives. (Joint work with M.Ronco.)

EMILY BURGUNDER (IMPAN, European Postdoctoral Institute)


17 November 2008 (Banach Center meeting GEOMETRY AND OPERATOR ALGEBRAS)

DEGENERATION OF NON-COMMUTATIVE COMPACT METRIC SPACES

Given a spectral triple associated to a unital C*-algebra and an extension of the C*-algebra by the compacts, we construct a 2-parameter family of spectral triples associated to the extended C*-algebra. In this way we obtain a two-parameter family of noncommutative compact metric spaces. By a variation of the parameters, we can obtain the compacts as well as the original C*-algebra as degeneration limits in the sense of noncommutative compact metric spaces. This is a joint work with Cristina Ivan, Hannover.

ERIK CHRISTENSEN (University of Copenhagen)


17 November 2008 (Banach Center meeting GEOMETRY AND OPERATOR ALGEBRAS)

APPLICATIONS OF THE CLASSIFICATION PROGRAM FOR C*-ALGEBRAS TO THE THEORY OF PERTURBATIONS OF C*-ALGEBRAS

The classification program provides results which tell that for certain classes of C*-algebras there is a complete set of invariants, such as K-groups, traces and the pairing of the traces with K_0. For the perturbation question, we consider two subalgebras of a common bigger C*-algebra, and we say that the algebras are close if their unit balls are close in the Hausdorff metric induced by the norm. One question is then if algebras that are sufficiently close are isomorphic. A way to a positive answer is to show that the invariants used in the classification results are stable under small perturbations of algebras. We give some positive answers to questions of this type. This is a joint work with Allan Sinclair, Edinburgh, Roger Smith, Texas, and Stuart White, Glasgow.

ERIK CHRISTENSEN (University of Copenhagen)


18 November 2008 (Banach Center meeting GEOMETRY AND OPERATOR ALGEBRAS)

FRACTALS STUDIED VIA NONCOMMUTATIVE GEOMETRY

A fractal set, such as the Cantor set or the Sierpinski gasket, is by no means smooth. Anyway, the theory developed to describe noncommuative smooth manifolds can be applied in this setting, and we can recover geodesic distances, Minkowski dimensions, Hausdorff measures and elements of K-homology in this way. The results are obtained in collaboration with Cristina Ivan, Hannover.

ERIK CHRISTENSEN (University of Copenhagen)


1 December 2008

CONFORMAL STRUCTURES IN NONCOMMUTATIVE GEOMETRY

It is well known that a compact Riemannian spin manifold (M,g) can be reconstructed from its canonical spectral triple (C^\infty(M),L^2(SM),D), where SM denotes the spinor bundle and D the Dirac operator. We show that the Riemannian metric g can be reconstructed up to conformal equivalence from (C^\infty(M),L^2(SM),sign(D)).

CHRISTIAN BÄR (Universität Potsdam)


8 December 2008

UNIQUENESS OF CUP PRODUCTS IN HOPF-CYCLIC COHOMOLOGY

Using the derived functor interpretation of the Hopf-cyclic cohomology with arbitrary coefficients, we will show that all cup products and pairings defined in the literature extending Connes-Moscovici characteristic map are either (i) identical if the computations are done in the same derived category, or (ii) are related via isomorphic double functors acting on equivalent derived categories. We will construct comparison natural transformations compatible with the pairings we are interested, and therefore conclude that these pairings are also isomorphic.

ATABEY KAYGUN (Max-Planck-Institut, Bonn, Germany)


15 December 2008

SPECTRAL ACTION ON NONCOMMUTATIVE TORI AND SUq(2)

I shall review the calculations of the spectral action for the noncommutative tori and the quantum group SUq(2). The results are obtained using the Chamseddine-Connes formula for the asymptotic expansion of the action. I'll discuss the similarities and differences between the classical (commutative) and the quantum (noncommutative) cases and speculate on the physical contents of the results.

ANDRZEJ SITARZ (Uniwersytet Jagielloński)


5 January 2009

L-INFINITY-STRUCTURE ON THE COMPLEX OF DEFORMATIONS OF MORPHISMS OF ALGEBRAS VIA A RESOLUTION OF OPERADS

We will explain how one can construct an L-infinity-structure on the complex of deformations of morphisms of algebras, which is an analog of the Gerstenhaber bracket for associative algebras. In particular, we can, with this tool, recover the Gerstenhaber-Schack cohomology complex and write a Maurer-Cartan equation whose solutions correspond to deformations of the morphisms. Our construction is based on a resolution of the coloured operad encoding morphisms. A possible application in non-commutative geometry could be to search for a definition of non-commutative versions of hamiltonian actions using this tool: a hamiltonian action can be viewed as a morphism of Lie algebras between the Lie algebra of symmetries and the Poisson algebra of functions on a Poisson manifold. Next, one can quantize this Poisson algebra by deformation to get a non-commutative associative algebra. On the other hand, a quantum group is a deformation of an envelopping algebra. Hence the idea would be to do simultaneously these two deformations, together with a deformation of the morphism, in order to get a "quantum moment map".

YAËL FRÉGIER (University of Luxembourg)


12 January 2009

LOCALIZED ENDOMORPHISMS OF THE CUNTZ ALGEBRAS

We discuss recent work on endomorphisms and automorphisms of the Cuntz algebras O_n that preserve both the core UHF-subalgebra and the canonical maximal abelian subalgebra. In particular, we present a new combinatorial approach to the study of such endomorphisms. We also briefly discuss connections with classical dynamical systems, index theory for subfactors, and noncommutative entropy. This talk is mainly based on a joint work with Roberto Conti and Jason Kimberley.

WOJCIECH SZYMAŃSKI (University of Southern Denmark, Odense)


19 January 2009

GALOIS AND ALGEBRAIC LATTICES

Galois methods found astonishing and wide usage in non-statistical data analysis and software engineering as the theoretical basis of Formal Concept Analysis. In many applications, one can present data as a formal concept: A set of objects, a set of attributes and a relation that says which attribute is possesed by a given object. It is a standard result (known already to Birkhoff) that one can associate to such a formal context (by means of a Galois aonnection) a complete lattice of formal concepts. However, only later on (in the paper by Wille in 1982), it was realized that this lattice is a very useful tool for data visualization and analysis. After an introduction to the research area, I will report on the paper "A categorical view on algebraic lattices in formal concept analysis", by P.Hitzler, M.Kroetzschand, and G.Zhang, which explores the notion of algebraicity in formal concept analysis from a categorical point of view.

BARTOSZ ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


16 February 2009

BIVARIANT K-THEORY VIA CORRESPONDENCES

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism.

RALF MEYER (Universität Göttingen, Germany)


23 February 2009

EXPANDERS, EXACT GROUPS, AND K-THEORY OF CROSSED-PRODUCT C* ALGEBRAS (PART 1)

These two talks will begin with the definition and basic properties of expander graphs. This will then be related to BCC (Baum-Connes conjecture with coefficients) via the observation that any group for which BCC is valid is K-theoretically exact. A key question is: Do non-exact groups exist? All the groups which mathematicians encounter in "real life" are exact. A finitely generated discrete group which contains in its Cayley graph a sub-graph which is an expander is not exact, and is not even K-theoretically exact, and is therefore a counter-example to BCC. Starting with a construction of Gromov, group theorists have tried to construct such a group. It seems quite possible that such a group exists, and the proof of its existence may have been recently completed.

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


2 March 2009

EXPANDERS, EXACT GROUPS, AND K-THEORY OF CROSSED-PRODUCT C* ALGEBRAS (PART 2)

These two talks will begin with the definition and basic properties of expander graphs. This will then be related to BCC (Baum-Connes conjecture with coefficients) via the observation that any group for which BCC is valid is K-theoretically exact. A key question is: Do non-exact groups exist? All the groups which mathematicians encounter in "real life" are exact. A finitely generated discrete group which contains in its Cayley graph a sub-graph which is an expander is not exact, and is not even K-theoretically exact, and is therefore a counter-example to BCC. Starting with a construction of Gromov, group theorists have tried to construct such a group. It seems quite possible that such a group exists, and the proof of its existence may have been recently completed.

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


9 March 2009

HOMOLOGICAL PRODUCTS AND DUALITY VIA HOPF x_A-ALGEBRAS

The aim of this joint work with Niels Kowalzig is twofold: on the one hand to unify the treatment of Poincare-type dualities arising from cup and cap products in algebraic (co)homology theories such as group, Lie algebra, Hochschild or Poisson (co)homology, and on the other hand to advertise Schauenburg's Hopf x_A-algebras as the algebraic structure on the relevant enveloping algebra that gives rise to these products.

ULRICH KRÄHMER (University of Glasgow, Scotland)


16 March 2009

SPECTRAL GEOMETRY OF CIRCLE BUNDLES

Taking as a starting point the analysis of Dirac operators on circle bundles (by Bär and Amman), we discuss the spectral triples arising in the case of the Hopf fibration (both in the commutative and in the quantum case).

ANDRZEJ SITARZ (Uniwersytet Jagielloński)


23 March 2009

ALGEBRAIC MODELS FOR EQUIVARIANT COHOMOLOGY OF NONCOMMUTATIVE SPACES

We study symmetries of noncomutative spaces, and the associated Hopf module algebra structures, focusing in particular on deformations involving Hopf algebra Drinfeld twists. We then define a deformed Weil algebra which we use to construct algebraic models for the equivariant cohomology of such actions. We finally show some basic properties of this twisted noncommutative equivariant cohomology, and discuss how to generalize the previous construction to Drinfeld-Jimbo deformations.

LUCIO CIRIO (Max-Planck-Institut, Bonn, Germany)


30 March 2009

FINITE CLOSED COVERINGS OF COMPACT QUANTUM SPACES

Coverings of spaces generate a distributive lattice of subspaces. In this vein, coverings of unital algebras (compact quantum spaces) are understood as sets of ideals that intersect to zero and generate a distributive lattice. We begin by observing that the affine covering of a complex projective space CP^n generates a free distributive lattice with n+1 generators. In order to obtain a noncommutative model for a free distributive lattice with n+1 generators, we construct a new quantum complex projective space QCP^n. (For n=1, our noncommutative projective space coincides with the mirror quantum sphere.) All this leads us to a Z/2 infinite projective space endowed with Alexandrov-type topology as a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this projective space. In technical terms, we prove that an appropriate category of finitely generated distributive lattices of ideals is equivalent to a category of finitely supported flabby sheaves of algebras. Finally, we explain how Maszczyk's global patterns (closed-set version of flabby sheaves) can be used for such a classification. (Joint work with Piotr M. Hajac and Chiara Pagani.)

ATABEY KAYGUN (University of Buenos Aires, Argentina) BARTOSZ ZIELIŃSKI (Uniwersytet Łódzki / IMPAN)


9 April 2009 (exceptional time: Thursday at 14:15)

TWO HOPF ALGEBRAS OF TREES INTERACTING

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. They are essentially characterized by the coproduct map. In this work we define yet another Hopf algebra H by introducing a new coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feyman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua. (Joint work with Kurusch Ebrahimi-Fard and Dominique Manchon.)

DAMIEN CALAQUE (Université Claude Bernard, Lyon, France)


20 April 2009

EQUIVARIANT K HOMOLOGY

In this talk a geometric model (along the lines of Baum-Douglas) for equivariant K homology will be given when the group G is a compact Lie group or a countable discrete group. The twisted version of this has been defined by Bai-Ling Wang and is tantamount to the D-branes of string theory. A recent development is the geometric realization of the Tate theory associated to equivariant K theory. The above is joint work with N.Higson, J.Morava, H.Oyono-Oyono, and T.Schick.

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


27 April 2009

COMPACT C*-QUANTUM GROUPOIDS

Quantum groupoids have successfully been axiomatised and studied in the finite case by Boehm, Szlachanyi, Nikshych, Vainerman and others, and in the measurable case by Enock, Lesieur and Vallin who were motivated by depth 2 inclusions of factors. In this talk, we define compact quantum groupoids in the setting of C*-algebras, study some of their properties, and discuss examples.

THOMAS TIMMERMANN (Universität Münster, Germany)


4 May 2009

ALGEBRAIC QUANTUM GROUPS

The road from Hopf algebras to locally compact quantum groups has been long, difficult, and with many obstacles. In the end, however, the result is a rich and very nice theory. In this talk I would like to 'walk' (or rather to 'drive') along this road and 'hang around' at certain places on the way. On this journey, I will in particular look at a few interesting aspects of the theory, like the antipode, the integrals and their modular properties in more detail. As an illustration of the various relations between all these objects, I will focus on Radford's formula giving the fourth power of the antipode in terms of the modular elements and discuss it at the different steps on the way. Moreover, I will make a longer stop when we encounter the algebraic quantum groups (i.e. the multiplier Hopf algebras with integrals) as they serve as a good model, purely algebraic in nature, for the far more difficult analytical theory of locally compact quantum groups.

ALFONS VAN DAELE (Katholieke Universiteit Leuven, Belgium)


11 May 2009

MORITA EQUIVALENCE OF ALGEBRAIC QUANTUM GROUPS

Let A be an algebraic quantum group and Y a unital right A-module. I introduce the notion of a coproduct on Y and give conditions for Y to be called a Morita A-module coalgebra. I will show that Y has a lot of structure, very similar to that of any algebraic quantum group. In particular, we have a counit, an antipode-like map, etc. We have integrals on Y, modular maps for these integrals, the modular element relating the left and the right integral, etc. It is possible to construct a reflected algebraic quantum group C, acting from the left on Y, making Y into a left Morita C-module coalgebra. And just as for algebraic quantum groups, one can construct the reduced dual Y^. It will be a bi-Galois object for the duals A^ and C^ of A and C, respectively. This is joint work with K. De Commer.

ALFONS VAN DAELE (Katholieke Universiteit Leuven, Belgium)


18 May 2009

DOCTRINAL HOPF-GALOIS EXTENSIONS

We will apply our idea of "quantum" spaces as (additive) monoidal categories, with doctrinal adjunctions (in the sense of Max Kelly) as geometric morphisms, as an alternative categorical framework for a noncommutative version of Hopf-Galois theory. This enables us to describe Hopf-Galois theory in terms extending its classical geometro-algebraic interpretation. In particular, we show the equivalences: 1) universal bundle = quotient stack of a point = Tannakian reconstruction through monoidal Eilenberg-Moore category, 2) affine structure of the universal quotient map = structure theorem for Hopf modules, 3) projection formula = antipode, 4) Beck-Chevalley condition for the classifying map = Hopf-Galois condition. Using Szlachanyi's characterization of Takeuchi's bialgebroids as monoidal comonads, we extend notions of the Hopf-Galois theory from Hopf algebras to appropriate Hopf bialgebroids. We will compare our doctrinal approach with other variants of Hopf bialgebroids existing in the literature.

TOMASZ MASZCZYK (Uniwersytet Warszawski / IMPAN)


25 May 2009

MONODROMY, TWISTING COCHAINS, AND CYCLIC CHERN CHARACTER

The notion of a cyclic Chern character (slighly different from the traditional one) was introduced by Bismut, Getzler, Jones and Petrack in late 1980-ies. This is a functorial construction that associates to every vector bundle over a manifold an element in 0-degree equivariant cohomology of the free loop space of the base. In my talk, I will try to explain, how this class can be related to such a purely algebraic concept as the twisting cochain associated with the bundle.

GEORGY SHARYGIN (ITEP, Moscow, Russia)


1 June 2009

NON-COMMUTATIVE INTEGRAL FORMS

The notion of a complex of integral forms on a non-commutative space is described. Following Manin, this is defined as a complex with a boundary operator given by a flat hom-connection (or a flat right connection in the terminology of Manin). We show how hom-connections, and therefore complexes of integral forms, can be constructed from free twisted multi-derivations. In particular, we show that a Hopf algebra with a left covariant differential calculus admits a hom-connection. Examples of integral forms include such forms on the matrix algebra (with the Lie algebra or derivation based calculus), on the quantum group SUq(2) (with the 3D calculus), and on the standard Podles sphere (with the calculus originated from the 3D calculus). In each case the complex of integral forms is shown to be isomorphic to the corresponding non-commutative de Rham complex, thus reflecting perfectly the classical case and also pointing in the direction of the Poincare duality. The constructed integrals coincide with the trace based integral of Dubois-Violette, Kerner and Madore (the matrix algebra case) and with the Haar integral (in the quantum group case). Based on joint work with Laiachi El Kaoutit and Christian Lomp.

TOMASZ BRZEZIŃSKI (Swansea University, G. Britain)