NONCOMMUTATIVE GEOMETRY SEMINAR
Mathematical Institute of the Polish Academy of Sciences
Ul. Śniadeckich 8, room 322, Mondays, 10:15-12:00
1999/2001
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2010/2011
10 October 2011
TOWARDS A QUANTUM MOMENT
MAP
Given a Poisson action of a Poisson-Lie group on a Poisson manifold, there is a well understood notion of the moment map and the associated Poisson reduction generalizing the classical Hamiltonian reduction of Marsden and Weinstein. The subject of this talk is the question of generalizing this to the case of quantum groups. We will describe what a good candidate for the "quantum moment map" is and give a few examples illustrating what happens with the reduction in this setting.
RYSZARD NEST
(Københavns Universitet, Denmark)
17 October 2011
NEW DIRAC OPERATORS THROUGH
NONCOMMUTATIVE CIRCLE BUNDLES
Motivated by classical circle bundles, we study spectral triples over the total space of noncommutative principal U(1)-bundles. We propose a compatibility condition between a connection and a Dirac operator, and analyze it in detail on noncommutative three-tori. Thus we find a family of new Dirac operators that arise from the base-space Dirac operator and a suitable connection. For noncommutative two-tori, we obtain a restricted version of the Gauss-Bonnet theorem. This is joint work with Ludwik Dabrowski.
ANDRZEJ SITARZ
(IMPAN/Uniwersytet Jagielloński)
3 November 2011 (Banach Center
research group.
Joint Noncommutative Geometry and
Operator Algebras
and Quantum Groups seminar. Exceptional time: Thursday 13:15.)
THE CUNTZ ALGEBRA Q_N AND C*-ALGEBRAS OF PRODUCT SYSTEMS
I will discuss a product system over
the multiplicative semigroup of positive integers of Hilbert bimodules
and show
how the associated Nica-Toeplitz algebra is related to the C*-algebra
Q_N introduced recently by Cuntz.
WOJCIECH SZYMAŃSKI
(Syddansk Universitet, Odense, Denmark)
7 November 2011 (Banach Center
research group.)
PRINCIPAL COMPACT QUANTUM GROUP ACTIONS ON UNITAL C*-ALGEBRAS ARE GALOIS
Let G be a compact group acting on a compact Hausdorff space.
Ellwood observed that one can easily
characterise the principality of this action on the level of the
associated C*-algebras by means of a density condition. More recently,
Baum and Hajac showed that Ellwood's condition is equivalent
to the principality of the associated Peter-Weyl comodule algebra,
and thus embedded the category of compact Hausdorff
principal bundles into the category of comodule algebras with Galois
coactions. It is the aim of this talk
to prove that the above notions are still equivalent when G
is a compact
quantum group acting on a unital C*-algebra.
(An important special case G=U(1) was proven much earlier by
W.Szymański.)
The key new element in our approach,
inspired by the work of S.Popa and A.Wassermann on ergodic compact group
actions on von Neumann algebras, is the use of Pimsner-Popa type
inequalities to allow passages between different Hilbert C*-modules
associated to an action of a compact quantum group. (Part of a joint
project with P.F.Baum, P.M.Hajac and W.Szymański.)
KENNY DE COMMER
(Université de Cergy-Pontoise, France)
7 November 2011 (Banach Center
research group.
Exceptional time: 14:15.)
EQUIVARIANT COMPARISON OF QUANTUM HOMOGENEOUS SPACES
We prove that quantum homogeneous spaces given by Poisson-Lie
quantum subgroups of the q-deformations of
simply connected simple compact Lie groups are equivariantly
KK-equivalent to the classical ones.
This result extends the nonequivariant case of Neshveyev and Tuset.
As an application,
we obtain an analogue of the Borsuk-Ulam theorem for quantum spheres
conjectured by Baum and Hajac.
MAKOTO YAMASHITA
(Università di Roma Tor Vergata, Italy)
14 November 2011
CONVERGENT SEQUENCES IN DISCRETE GROUPS AND CHU DUALITY
It is well known that a locally compact group can be recovered
from the category of its unitary representations equipped with the natural
operations of sum and tensor product. I will explain that a similar
approach fails when restricted to finite-dimensional representations,
unless the group is virtually abelian. In particular, even for a
non-abelian free group (which is residually finite), the natural bi-dual
group build on finite-dimensional representations is strictly larger than
the free group. This solves an old problem due to Hsin Chu and the methods
answer several questions about the Bohr topology of the free group.
ANDREAS THOM
(Universität Leipzig, Germany)
14 November 2011 (Exceptional time: 14:15.)
ALGEBRAIC GEOMETRY OF
TOPOLOGICAL SPACES
We use techniques from both real and complex algebraic geometry
to study K-theoretic and related invariants of the algebra C(X) of
continuous complex-valued functions on a compact Hausdorff topological
space X.
For example, we prove a parameterized version of a theorem of
Joseph Gubeladze, that is, we show that if M is a countable, abelian, cancellative,
torsion-free, seminormal monoid, and X is contractible, then every
finitely generated projective module over C(X)[M] is free. The particular
case when M=N^n
gives a parameterized version of the celebrated theorem,
proved independently by Daniel Quillen and Andrei Suslin, that all
finitely
generated projective modules over a polynomial ring over a field are free.
The conjecture of Jonathan Rosenberg which predicts the homotopy
invariance of the negative algebraic K-theory of C(X) follows from the
particular case when M=Z^n.
ANDREAS THOM
(Universität Leipzig, Germany)
21 November 2011
NONEXISTENCE OF NONCLASSICAL QUANTUM PERMUTATIONS ACTING ON MAPS
The quantum permutation group was introduced by S. Wang as
the universal compact quantum group acting on a finite set X
in a way preserving the counting measure of X.
In the classical case,
the counting measure can be interpreted as a function on the set
P(X)
of all subsets of X that is invariant under the action of the
permutation group on P(X) induced from its action on X.
More generally, the permutations of X
naturally act on Map(X,Y) for any set Y.
In particular, for Y={0,1} one
obtains the action of permutations on P(X).
However, we will show that quantum permutations of X
inducing a quantum action on spaces of maps
Map(X,Y)
must be necessarily classical. On the other hand, allowing
"quantum maps" acted on by quantum permutations one faces the fact
that classically X = Map({0},X)
is itself a set of maps. Therefore,
X itself should also be
quantized in an appropriate sense going beyond the context of
Wang's construction. This would generalize the concept of a quantum
family of maps studied by P.M. Sołtan.
TOMASZ MASZCZYK
(Uniwersytet Warszawski / IMPAN)
19 December 2011
HOPF IMAGES, IDEMPOTENT STATES AND MATRIX MODELS OF QUANTUM GROUPS
A Hopf image of a given representation of a Hopf algebra A is the largest Hopf quotient of A through which the representation factorises in a natural way. This notion was introduced by T. Banica and J. Bichon in 2010. If the Hopf image is equal to A, the considered representation is called inner faithful. The question of the existence of an inner faithful finite-dimensional representation of a given Hopf algebra is a natural counterpart of the investigation of linearity of a given discrete group G. Hence Hopf algebras admitting such representations are called inner linear. In this talk, we will recall the theory developed by Banica and Bichon, present a new approach to Hopf images of compact quantum groups via the theory of idempotent states developed mainly by U. Franz and A. Skalski, formulate some open problems related to inner linearity, and discuss connections to various notions of matrix models of quantum groups. (Joint work with Teodor Banica and Uwe Franz.)
ADAM SKALSKI
(IMPAN)
19 December 2011 (Exceptional time: 14:15.)
A CRITERION FOR STABLE TRIVIALITY OF THE ASSOCIATED NONCOMMUTATIVE LINE BUNDLES
A general criterion for the stable freeness of the finitely generated projective modules associated to principal comodule algebras via one-dimensional corepresentations will be proved. The line bundles over quantum real and complex projective spaces of an arbitrary dimension n will serve as examples. Here the criterion allows one to conclude that the stable non-freeness in the general case follows from the stable non-freeness for n=2 and n=1, respectively.
The real case for n=2 was handled by direct K-theory methods, and
the complex case for n=1 by the noncommutative index pairing.
In both cases the line bundles associated by the
non-trivial (non-zero winding number) corepresenations were shown to be not stably free. Hence all such
non-zero winding number line bundles over these quantum projective spaces are not stably free.
PIOTR M. HAJAC
(IMPAN/Uniwersytet Warszawski)
9 January 2012
CLOSED QUANTUM SUBGROUPS
The notion of a closed quantum subgroup of a given locally compact quantum group was introduced by S. Vaes. His definition uses a mixture of the C*-reduced, C*-universal, and von Neumann algebraic version of a given quantum group and its dual. Recently, S.L. Woronowicz proposed a definition based on the notion of a bicharacter and the concept of the C*-algebra generated by a quantum family of elements. In my talk, I shall give a number of equivalent characterizations of Woronowicz's definition and make a link between the Vaes and Woronowicz approaches.
(Joint work with Matthew Daws, Adam Skalski and Piotr M. Sołtan.)
PAWEŁ Ł. KASPRZAK
(Uniwersytet Warszawski/IMPAN)
9 January 2012 (Exceptional time: 14:15.)
REPRESENTATIONS OF GROUPOIDS AND IMPRIMITIVITY SYSTEMS
.
We study the correspondence between
the unitary representations of transformation groupoids and systems of
imprimitivity. Next, for the general case of locally
compact transitive groupoids, we define representations induced by a
representation of the isotropy subgroupoid and prove an imprimitivity
theorem. These results generalize classical concepts of Mackey known in the
representation theory of locally compact groups.
LESZEK PYSIAK
(Politechnika Warszawska)
16 January 2012 (Banach Center
research group.)
THE K-THEORY OF FREE QUANTUM
GROUPS
This talk concerns the K-theory of free quantum groups in the sense of Wang and Van Daele. More precisely, we show that the free products of free unitary and free orthogonal quantum groups are K-amenable, and establish an analogue of the Pimsner-Voiculescu exact sequence. As a particular consequence, we obtain an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of Baum-Connes conjecture methods to the framework of discrete quantum groups. It is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result, we show that the gamma-element of any free quantum groups equals 1. As an important ingredient in the proof, we adapt the Dirac-dual-Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.
(Joint work with Roland Vergnioux.)
CHRISTIAN VOIGT
(Universität Münster, Germany)
16 January 2012 (Banach Center
research group.
Exceptional time: 14:15.)
INVARIANTS FOR A
CONFORMALLY REGULAR PENTAGONAL TILING OF THE PLANE
The Bowers and Stephenson conformally regular pentagonal tiling of the plane enjoys remarkable combinatorial and geometric properties. Since it
does not have finite local complexity in any usual sense, it is beyond the standard tiling theory. On the other hand, the tiling can be completely
described by its combinatorial data that, rather automatically, has finite local complexity. With the aim to compute its K-theory, we construct the hull and C*-algebra of this tiling solely
from its combinatorial data.
As the tiling possesses no natural R^2 action by translation, there
is no a priori reason to expect that the K-theory of the C*-algebra of the tiling
is the same as the K-theory or cohomology of the hull of the tiling, and it would be very
interesting if they were different.
MARIA RAMIREZ-SOLANO
(Københavns Universitet, Denmark)
27 February 2012
EXACT CROSSED-PRODUCTS
AND A COUNTER-EXAMPLE (TO BAUM-CONNES WITH COEFFICIENTS)
REVISITED
Let G be a locally compact Hausdorff topological group which is second
countable. Let G-C* denote the category of all G-C* algebras.
The morphisms in G-C* are *-homomorphisms which are G-equivariant.
Consider crossed-products which are intermediate between the max
crossed-product and the reduced crossed-product. Such a crossed-product
is said to be exact if and only if, whenever
0 -> A -> B -> C -> 0 is a short exact sequence in G-C*, the sequence of C* algebras obtained
by applying the crossed-product is exact. For example, the max crossed
product is exact for any G. The reduced crossed-product is exact if and
only if G is exact. P. Baum conjectured and
E. Kirchberg proved that there always exists a unique minimal exact
crossed-product. The Baum-Connes (BC) conjecture with coefficients
should then be
reformulated to state that the BC map K_*(E_G,A) -> K_*(C*(G,A)) is an
isomorphism, where C*(G,A) is the minimal exact crossed-product. In this
reformulated
version of BC with coefficients it is probable that the
Higson-Lafforgue-Skandalis counter-example is no longer a
counter-example.
PAUL F. BAUM
(Pennsylvania State University, State College, USA / IMPAN)
5 March 2012 (Banach Center
research group.)
A FINITENESS CONDITION FOR COMPACT QUANTUM GROUP ACTIONS
By a theorem due to Hoegh-Krohn, Landstad and Stormer, the
isotypical components of an ergodic action of a compact group on a
unital C*-algebra are finite dimensional. This was later generalized to
compact quantum groups by F. Boca. On the other hand, since recently we know that the isotypical
components of a free action of a compact quantum group on a unital
C*-algebra are finitely-generated modules over the fixed-point subalgebra.
In
this talk, we give several characterizations, in terms of Galois maps,
of actions for which such a finite generation property holds in
the general setting of compact quantum groups acting on unital
C*-algebras. In the classical case, they are precisely the actions
whose field of isotropy groups is continuous. This is joint work with M.
Yamashita, and a joint project with P.F. Baum and P.M. Hajac.
KENNY DE COMMER
(Université de Cergy-Pontoise, France)
5 March 2012 (Banach Center
research group.
Exceptional time: 14:15.)
GROUP-DUAL SUBGROUPS OF COMPACT QUANTUM GROUPS
There are two special classes of compact quantum groups,
namely the classical compact groups and the
Fourier duals of discrete groups. In this talk, we discuss what can be
said about the discrete group duals sitting inside a given compact
quantum group. We will consider several examples such as Wang's quantum
automorphism groups and Goswami's quantum isometry groups of connected
compact Riemannian manifolds. This is joint work with T. Banica and J.
Bhowmick.
KENNY DE COMMER
(Université de Cergy-Pontoise, France)
12 March 2012
CONSTRUCTING SPECTRAL TRIPLES ON C*-ALGEBRAS
Spectral triples are a refinement of K-homology cycles modelled
on the Dirac operator of a compact spin manifold. There are various
regularity properties of spectral triples connected to dimension and
Rieffel's notion of a noncommutative or quantum metric space. We are mostly
interested in existence results for spectral triples with good properties,
e.g. defining good metrics on state spaces, which may be regarded as
noncommutative metrisation results. We will review some old and new
results and constructions of spectral triples especially on crossed
products.
This is joint work with Andrew Hawkins, Adam Skalski and Stuart White.
JOACHIM ZACHARIAS (The University of Nottingham, England)
12 March 2012 (Exceptional time: 14:15.)
CATEGORIES OF COLOURED PARTITIONS RELATED TO
REPRESENTATIONS OF COMPACT QUANTUM GROUPS
Although the main part of this talk will
have purely combinatorial/free-probabilistic nature, links
to quantum groups will
be described towards the end of the presentation.
We will discuss certain combinatorial problems related
to particular categories of partitions.
Non-crossing partitions
are fundamental combinatorial tools of free probability. In this
talk,
we will discuss their extended, so-called "coloured", versions.
These naturally arise
in relation to intertwiners of some unitary matrices. They
in turn are related to the representation theory of some
compact quantum symmetry groups. (Most presented
results come from joint work with Teo Banica.)
ADAM SKALSKI
(IMPAN)
19 March 2012
QUANTUM PERMUTATIONS OF TWO ELEMENTS
As discovered by Shuzhou Wang, the universal action of a compact quantum
group on the set of n elements non-trivially extends the
action of the classical permutation group beginning at n=4.
This fact has been generalized from compact quantum groups to cosemisimple or involutive Hopf algebras by Julien Bichon. Now we show that for the wider class of Hopf algebras with bijective antipode
non-classical
quantum permutations start to appear already at n=2. It is also the case
for the class of all Hopf algebras because
the universal property then has a solution in terms of Manin's Hopf envelope. However, unlike Manin's construction, the Hopf algebra admitting a coaction (on the algebra of functions on the set of n elements) that is universal in the class of Hopf algebras with bijective antipode
is presented as a localization of a finitely generated algebra.
TOMASZ MASZCZYK
(Uniwersytet Warszawski / IMPAN)
26 March 2012
KADISON-KASTLER STABILITY FOR OPERATOR ALGEBRAS
Kadison and Kastler equipped the set of all C*-subalgebras
of B(H) with a natural metric and conjectured that sufficiently close
algebras should be isomorphic. This is a uniform metric: two algebras
A and B are close in the Kadison-Kastler metric if every operator in
the unit ball of A can be well approximated in the unit ball of
B and
vice versa. Kadison and Kastler's conjecture was established in the
70's when one algebra is an injective von Neumann algebra. In this
talk, I will discuss some recent progress for non-injective von
Neumann algebras (mainly arising as crossed products) and the
connections between this problem and the similarity problem.
STUART WHITE
(The University of Glasgow, Scotland)
26 March 2012 (Exceptional time: 14:15.)
QUANTUM GROUPS AND SPECIAL FUNCTIONS
Since the 1950ies there has been a close connection between special
functions and Lie groups, and this cross fertilization has turned out to
be very fruitful for both sides. Even today this relation is a source of
new and interesting results.
Several aspects of this interplay found their generalization to
appropriate relations between quantum groups and special functions of
basic hypergeometric type almost immediately after the introduction of
quantum groups at the end of the 1980ies. Already compact quantum
groups led to new and interesting results for special functions,
including, e.g.,
addition and product formulae which would have been hard to
obtain otherwise.
Later, special functions were used in the construction of certain
non-compact quantum groups. In particular, this concerns the quantum
analogue of the normalizer of SU(1,1) in SL(2,C).
The harmonic analysis on this quantum
group has given rise to some new notions and results for the associated
special functions.
In this talk, the general development of the relation between compact
quantum groups and special functions will be shortly reviewed.
Then we extend
the picture to include non-compact quantum groups. In particular, we
will consider the aforementioned example
of a quantum-group normalizer and its relation to special functions.
We will end by discussing the status of a personal wish-list of
results we desire for the interpretation of special functions on quantum
groups.
ERIK KOELINK
(Radboud Universiteit Nijmegen, the Netherlands)
2 April 2012
THE K-THEORY OF NONCOMMUTATIVE BIEBERBACH SPACES
Bieberbach manifolds are compact quotients of R^n
by a free, properly
discontinous
and isometric action of a discrete group. The noncommutative counterparts
of Bieberbach spaces in dimension 3 arise as quotients of a
three-dimensional noncommutative torus by an action of a finite group. We
compute the K-theory of all such
noncommutative Bieberbach spaces.
Our methods use the results (of Walters and Echterhoff et al)
on unbounded twisted traces and on an explicit
presentation of the K-theory generators of the quotients of a
two-dimensional noncommutative torus by an action of a cyclic group.
PIOTR OLCZYKOWSKI
(Uniwersytet Jagielloński)
23 April 2012
THE DUALITY OF GENERALIZED
HOPF AND LIE ALGEBRAS
Let H be a Hopf algebra and P be the functor that assigns to each Hopf algebra the Lie algebra of its primitive elements.
Michaelis introduced the notion of a Lie coalgebra and
defined a dual functor Q that assigns to each Hopf algebra a Lie coalgebra.
Moreover, he proved that P of the Sweedler dual of H
is isomorphic with the space of functionals on Q(H).
The aim of this talk is to show a more abstract version of the Michaelis
theorem, so that it can be applied to a broader class of Hopf algebra
type objects and dualities.
(Joint work with Isar Goyvaerts.)
JOOST VERCRUYSSE
(Université Libre de Bruxelles, Belgium)
23 April 2012 (Exceptional time: 14:15.)
CLEFT-TYPE BICOMODULES AND
GALOIS CO-OBJECTS
Let B be a k-bialgebra. The "fundamental theorem for Hopf modules" can be stated as the
equivalence of the following three properties:
(i) B admits an antipode (i.e. B is a Hopf algebra);
(ii) the canonical map for the coproduct coaction of B on itself is bijective; (iii) the category of
k-modules is equivalent to the category of (right, right)
B-Hopf
modules through the functor of tensoring with B on the right.
The three properties have been generalized in various ways and settings
not necessarily preserving the equivalence between the properties (e.g.,
in the setting of corings and entwining structures). For
instance, a "cleft" bicomodule is defined by means of a cleaving map
that mimics the behaviour of the antipode, a "Galois comodule" has a
bijective canonical map, and a "Galois co-object" always induces an
equivalence of categories between k-modules and a generalized category
of Hopf-modules.
Our aim is to establish the equivalence of similar three
conditions in the setting of (dual) Galois comodules. In particular, this
will allow us to show that Galois co-objects are equipped with an analogue
of the antipode map.
JOOST VERCRUYSSE
(Université Libre de Bruxelles, Belgium)
7 May 2012 (Banach Center
research group.)
NONCOMMUTATIVE
DIFFERENTIAL GEOMETRY OF A QUANTUM
DEFORMATION OF THE 7-DIMENSIONAL HOPF FIBRATION
For well over a decade it has been an open problem to find a
q-deformed
analogue of the principal SU(2) Hopf fibration of the 7-dimensional sphere equipped with
finite-dimensional noncommutative differential structures. In the
literature one finds a range of examples of such quantum `quaternionic'
Hopf fibrations, but only at the level of infinite dimensional `universal'
differential structures.
In this talk I will present an example of a q-deformed quaternionic Hopf
fibration whose total space and base space
are equipped with finite-dimensional differential structures. By
investigating the quantum symmetries of the fibration, I will describe the
geometry of the corresponding quantum twistor space (i.e.
a noncommutative 3-dimensional
complex projective space) and
use it to study a system of anti-self-duality equations on
the base space quantum
4-dimensional sphere. The system of equations admits a canonical `instanton' solution coming from a natural
projection defining a noncommutative vector bundle over this quantum sphere.
(Joint work with G. Landi.)
SIMON BRAIN
(Université du Luxembourg)
7 May 2012 (Banach Center
research group.
Exceptional time: 14:15.)
A NON-COMODULE-ALGEBRA EXAMPLE OF THE PULLBACK OF PRINCIPAL COACTIONS
The class of principal coactions is closed under one-surjective pullbacks
in an appropriate category of algebras equipped with left and right coactions. This allows us to go beyond the
category of comodule algebras when constructing examples of principal
coactions. The aim
of this talk is to show such an example by constructing a family of coalgebraic noncommutative deformations of the
U(1)-principal bundle S7 → CP3.
(Joint work with P. M. Hajac.)
ELMAR WAGNER
(Universidad Michoacana de San Nicolás de Hidalgo,
Morelia, Mexico)
28 May 2012
THE BAAJ-SKANDALIS
"ax+b" QUANTUM GROUP AS AN EXAMPLE OF A C*-ALGEBRAIC DRINFELD
TWIST
For a Lie group G and three closed subgroups A, B,
C, such that the intersection of A with B and
C
with B is trivial, G=AB and the intersection of BC
with CB is open and dense in G, we
construct a
Drinfeld twist that can be used to twist the
comultiplication of a certain quantum
group related to the decomposition G=AB. We exemplify this construction by
the Baaj and Skandalis "ax+b" quantum group and, time permitting,
by the kappa-Poincare quantum group.
PIOTR STACHURA
(Warsaw University of Life Sciences)
28 May 2012 (Exceptional time: 14:15.)
NONCOMMUTATIVE
CORRESPONDANCES AND APPLICATIONS TO GAUGE THEORY
We will discuss how KK-theory can be constructed via correspondences of
spectral triples. This involves the notion of a connection analogous to a
vector bundle connection. By factoring a spectral triple into a
correspondence applied to a spectral triple over a commutative base, we
can describe the gauge theory of the initial spectral triple in terms of
connections. We will exemplify this construction on noncommutive tori and
quantum Hopf fibrations. (Joint work with S. Brain.)
BRAM MESLAND
(The University of Manchester, England)