
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR 
Focus
Program on Noncommutative Geometry and Quantum Groups
LECTURES
AND ACTIVITIES
June
1014,
2013
Theme Week on Quantum Groups and Hopf Cyclic Homology
Location: Bahen Building, Room 1190 (map)
Organized by Piotr M. Hajac


Preliminary Schedule:
Monday, June 10

9:3010:30

Minicourse: Bahram Rangipour (University
of New Brunswick)
Motivation and preliminaries 
10:3011:00

Break 
11:00  12:00

Seminar Session: Mohammad Hassanzadeh (University
of Windsor )
Generalized Coefficients for Hopf cyclic cohomology

2:00  3:00

Minicourse: Alfons Van Daele(University
of Leuven  Belgium)
The Haar weights on a locally compact quantum
group (lecture slides)
Lecture references included in the lectures: Paper
1, paper 2 
3:00  3:30

Tea Break 
3:30  4:30

Seminar Session: Alfons Van Daele (University of
Leuven  Belgium)
Weak multiplier Hopf algebras versus
Multiplier Hopf Algebroids (lecture
slides) 
Tuesday, June 11

9:3010:30

Minicourse: Alfons Van Daele (University
of Leuven  Belgium)
The antipode of a locally compact quantum group
(lecture slides) 
10:3011:00

Break 
11:00  12:00

Seminar Session: Kenny De Commer (Université
de CergyPontoise)
Quantization of coadjoint orbits 
2:00  3:00

Minicourse: Bahram Rangipour (University
of New Brunswick)
Construction and examples 
3:00  3:30

Tea Break 
3:30  4:30

Seminar Session: Tomasz Maszczyk (University of Warsaw
)
Hopfcyclic theory of monoidal functors 
Wednesday, June 12

9:3010:30

Minicourse: Bahram Rangipour (University
of New Brunswick)
ConnesMoscovici type Hopf algebras 
10:3011:00

Break 
11:00  12:00

Seminar Session: Giovanni Landi (Universita
di Trieste)
The Weil algebra of a Hopf algebra 
2:00  3:00

Minicourse: Alfons Van Daele (University
of Leuven  Belgium)
The main results about locally compact quantum
groups (lecture slides) 
3:00  3:30

Tea Break 
3:30  4:30

Seminar Session: ByungJay Kahng (Canisius College)
Definition of a quantum groupoid in the C*algebra
framework 
Thursday, June 13

9:3010:30

Minicourse: Alfons Van Daele (University
of Leuven  Belgium)
The dual of a locally compact quantum group
(lecture slides) 
10:3011:00

Break 
11:00  12:00

Seminar Session: Makoto Yamashita (Ochanomizu
University)
Classification of quantum homogeneous spaces 
2:00  3:00

Minicourse: Bahram Rangipour (University
of New Brunswick)
Computational methods 
3:00  3:30

Tea Break 
3:30  4:30

Seminar Session: Serkan Sutlu (University
of New Brunswick)
On the characteristic classes of symplectic foliations
via Hopfcyclic cocycles 
Friday, June 14

9:3010:30

Minicourse: Bahram Rangipour (University
of New Brunswick)
Characteristic maps and cup products 
10:3011:00

Break 
11:00  12:00

Seminar Session: Ulrich Kraehmer (University of Glasgow
)
New remarks on the Dirac operators on quantised
Hermitian symmetric spaces 
2:00  3:00

Minicourse: Alfons Van Daele (University
of Leuven  Belgium)
Special cases, examples and generalizations
(lecture slides) 
3:00  3:30

Tea Break 
3:30  4:30

Seminar Session: Shuzhou Wang (University
of Georgia)
Equivalent Notions of Normal Quantum Subgroups, Compact
Quantum Groups with Properties F and FD, and Other Applications 
5:006:00

Seminar Session: Greg Maloney (University
of Toronto)
The ultrasimplicial property for rank two simple dimension
groups with unique state, the image of which has rank one 
Seminars:
Speaker 
Title and Abstract 
Tomasz Maszczyk
University of Warsaw 
Hopfcyclic theory of monoidal functors.
We present a categorical approach offering a conceptual simplification
of Hopfcyclic theory. The main point consists in appropriate identification
of the role played by different levels of the hierarchy consisting
of objects, morphisms, functors, natural transformations and the monoidal
structure in the context of the
concrete realization of Hopfcyclic theory based on algebras, coalgebras,
Hopf bialgebroids and coefficients in stable antiYetterDrinfeld
modules. In particular, algebras are replaced by monoidal functors,
stable antiYetterDrinfeld modules of coefficients by some other
functors and traces by some natural transformations. The classical
version is merely a component corresponding to the monoidal unit.
This approach is different from other categorifications as that of
BohmStefan or that of Kaledin.

ByungJay Kahng Canisius
College 
Definition of a quantum groupoid in the C*algebra framework
In this talk, I will report on my recent (and ongoing) joint work
with Alfons Van Daele on developing the definition of a locally compact
(C*algebraic) quantum groupoid. At the purely algebraic level, this
is closely related with the notion of ``weak multiplier Hopf algebras''
by Van Daele and Wang. Here, the comultiplication map cannot be nondegenerate,
so a special idempotent element E plays an important role.
Many of the techniques from the locally compact quantum group theory
carry over, with some adjustments. However, there are some different
challenges, concerning the canonical idempotent E (which would be
1 in the quantum group case), working with the Haar weights, and the
like. As time permits, I will give some explanations on these aspects,
as well as some possible future applications. The work is still ongoing,
but the expected finished version should be closely related with the
notion of ``measured quantum groupoids'' by Enock, Lesieur, and others,
while technically somewhat different.

Makoto Yamashita
Ochanomizu University 
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 particularly
nice classification in terms of the category of equivariant Hilbert
modules. In the case of quantum SU(2), we obtain a simple description
of equivariant homomorphisms and Kgroups in terms of certain weighted
graphs, based on the categorical structure of such modules. Based on
joint work with Kenny De Commer.

Giovanni Landi
Universita di Trieste 
The Weil algebra of a Hopf algebra
We generalize the notion, due to H. Cartan, of an operation of a
Lie algebra in a graded differential algebra. Firstly, for such an
operation we give a natural extension to the universal enveloping
algebra of the Lie algebra and analyze all of its properties. Building
on this we define the notion of an Hoperation, thai is the operation
of a general Hopf algebra H in a graded differential algebra. We then
introduce for such an operation the notion of algebraic connection.
Finally we discuss the corresponding noncommutative version of the
Weil algebra as the universal initial object of the category of Hoperations
with connections.

Kenny De Commer
Université de CergyPontoise 
Quantization of coadjoint orbits
We present an operator algebraic approach to the quantization of
certain coadjoint orbits, with modified Poisson bracket, in the PoissonLie
duals of semisimple compact Lie groups.

Mohammad Hassanzadeh
University of Windsor 
Generalized Coefficients for Hopf cyclic cohomology
In this talk, we introduce the category of coefficients for
Hopf cyclic cohomology. We see that this category has two proper subcategories
whose the smallest one is the known category of stable anti YetterDrinfeld
modules. The middle subcategory comprised of those coefficients which
satisfy a generalized SAYD condition depending on both the Hopf algebra
and the (co)algebra in question. We introduce several examples to show
that these three categories are different. This is a joint work with
Bahram Rangipour and
Dan Kucerovsky.

Ulrich Kraehmer
University of Glasgow 
New remarks on the Dirac operators on quantised Hermitian symmetric
spaces
In this joint work with Matthew TuckerSimmons (U Berkeley) the Dolbeault
complex of the quantised compact Hermitian symmetric spaces is identified
with the Koszul complexes of the quantised symmetric algebras of Berenstein
and Zwicknagl, which then leads to an explicit construction of the
relevant quantised Clifford algebras.

Greg Maloney University
of Toronto 
The ultrasimplicial property
for rank two simple dimension groups with unique state, the image of which
has rank one
In the 1970s, Elliott gave a complete classification of AF C*algebras
using the K_0 functor. The ordered groups in the range of this invariant
are precisely the countable members of the class of dimension groups.
One question that has remained unanswered since the early days of
this theory is how to characterize those countable dimension groups
that are ultrasimplicial, meaning that they can be written as inductive
limits of simplicial groups in which the maps are injective.
I will present some positive and negative results for rank two simple
dimension groups with unique state, the image of which has rank one.

Serkan Sutlu
University of New Brunswick 
On the characteristic classes of symplectic foliations via Hopfcyclic
cocycles
We report on our ongoing joint work with B. Rangipour on
realizing the characteristic classes of symplectic foliations of codimension
2 in the cyclic cohomology of the groupoid action algebra.
Using a cup product construction, one can construct a characteristic
homomorphism from the truncated Weil algebra of the general linear
Lie algebra gl(n) to the groupoid action algebra upon which the ConnesMoscovici
Hopf algebra H_n acts.
By this characteristic homomorphism, we transfer the characteristic
classes of (smooth) foliations of codimension 1 and 2 to the cyclic
cohomology of the groupoid action algebra, recovering the results
of Connes and Moscovici in codimension 1.
One then expects to carry out a similar construction, with the symplectic
Hopfalgebra SpH_n, to transfer the characteristic classes of symplectic
foliations. In this talk, we briefly disscuss the challenges/problems
we have encountered in this symplectic case.

Alfons Van Daele
University of Leuven  Belgium 
Weak multiplier Hopf algebras versus Multiplier Hopf Algebroids
For any group ${G}$, the algebra ${K(G)}$ of complex functions on
${G}$ (with pointwise operations) is a multiplier Hopf algebra when
the coproduct ${\Delta}$ on ${K(G)}$ is defined as usual by ${\Delta(f)(p,q)=
f(pq)}$ when ${p,q \in G}$. If ${G}$ is not a group, but only a groupoid,
it is still possible to define a coproduct as above, provided we let
${\Delta(f)(p,q)= 0}$ when ${pq}$ is not defined. Then ${(K(G),\Delta)}$
is a weak multiplier Hopf algebra. Recall that the difference between
a Hopf algebra and a weak Hopf algebra lies in the fact that the coproduct
is no longer assumed to be unital. The difference between a multiplier
Hopf algebra and a weak multiplier Hopf algebra is similar.
Weak multiplier Hopf algebras can be considered as quantum groupoids.
However, in some sense, the theory is too restrictive. A more adequate
notion is that of a multiplier Hopf algebroid. Roughly speaking, in
the case of a multiplier Hopf algebroid, it is no longer assumed that
the base algebra is separable (in the sense of ring theory). Recall
that the base algebra in the case of a groupoid as above, is the algebra
of complex functions with finite support on the set of units of the
groupoid. In this case, the base algebra is always separable, but
this need not be so for general quantum groupoids.
In this talk, I plan to discuss the relation of these two notions,
weak multiplier Hopf algebras and multiplier Hopf algebroids. Of course,
I will first give precise definitions of the two concepts. A simple
example will be given to explain the difference. At the end of the
talk I will say something about the importance of this difference
for the more involved theory of locally compact quantum groupoids
and the relation with the work on measured quantum groupoid (as studied
by Enock, Lesieur and others).
This is about work in progress with Shuanhong Wang (Southeast University
of Nanjing  China), Thomas Timmermann (University of M${\ddot{u}}$nster
 Germany) and ByungJay Kahng (Canisius College Buffalo  USA).

Shuzhou Wang
University of Georgia 
Equivalent Notions of Normal Quantum Subgroups, Compact Quantum
Groups with Properties F and FD, and Other Applications
In purely algebraic context, Parshall and Wang introduced a notion
of normal quantum subgroups for all Hopf algebras using adjoint coactions.
In the setting of compact quantum groups, I introduced another notion
of normal quantum subgroups using representation theory. I will first
show that these two notions of normality are equivalent for compact
quantum groups. As applications, I will introduce a quantum analog
of the third fundamental isomorphism theorem for groups, which is
used along with the equivalence theorem to obtain results on structure
of quantum groups with property F and quantum groups with property
FD. Other results on normal quantum subgroups for tensor products,
free products and crossed products will also be introduced if time
permits.

MiniCourse: 
Alfons Van Daele
University of Leuven  Belgium 
Locally compact quantum groups
To any locally compact group ${G}$, one can associate two
${C^*}$algebras. First there is the algebra ${C_0(G)}$ of complex functions
on ${G}$ tending to 0 at infinity. Next, we have the reduced ${C^*}$algebra
${C_r^*(G)}$. There is a natural duality between the two in such a way
that the product on one component induces a coproduct on the other one.
The coproduct ${\Delta}$ on ${C_0(G)}$ is given by the formula ${\Delta
(f)(p, q) = f(pq)}$ whenever ${p, q \in G}$. The coproduct ${\Delta}$
on ${C_r^*(G)}$ is characterized by ${\Delta(\lambda_p) = \lambda_p
\bigotimes \lambda_p}$ where p ${\mapsto \lambda_p}$ is the canonical
imbedding of ${G}$ in the multiplier algebra of ${C_r^* (G)}$. The above
duality generalizes the Pontryagin duality for locally compact abelian
groups to the nonabelian case.
In the theory of locally compact quantum groups, the idea is to 'quantize'
the above system.The abelian ${C^*}$algebra ${C_0(G)}$ is replaced
by any ${C^*}$algebra ${A}$ and the coproduct ${\Delta}$ on ${C_0(G)}$,
induced by the product in ${G}$, is replaced by any coproduct ${\Delta}$
on the ${C^*}$algebra A. Further assumptions on the pair ${(A,\Delta)}$
are necessary for it to be called a locally compact quantum group. Then
the dual can be constructed and it is again a locally compact quantum
group.
In the general theory, unfortunately, the existence of the quantum analogues
of the Haar measures, the Haar weights on the pair ${(A,\Delta)}$, has
to be assumed. On the other hand, as it turns out, these Haar weights
are unique, if they exist. And moreover, in examples, there are most
of the time obvious candidates for which it is not difficult to prove
that they satisfy the requirements.
There is also a formulation of the theory in the setting of von Neumann
algebras. This is not so natural, from a philosophical point of view,
but on the other hand, it seems to allow an easier treatment. And as
the two approaches are completely equivalent and in the end yield the
same objects, we will follow the more easy von Neumann algebraic track
to develop the theory. Still we will explain how this can be used to
understand the ${C^*}$approach as well.
Content of the five lectures:
1. The Haar weights on a locally compact quantum group
2. The antipode of a locally compact quantum group
3. The main results about locally compact quantum groups
4. The dual of a locally compact quantum group
5. Special cases, examples and generalizations
In the first lecture, we will need to review the basics of the theory
of lower semicontinuous weights on ${C^*}$algebras and normal weights
on von Neumann algebras, in relation with various aspects of the TomitaTakesaki
theory. In the middle three lectures, we will develop the theory. And
in the last lecture, if time permits, we will also say something about
the various directions of recent developments.

Bahram Rangipour
University of New Brunswick 
HopfCyclic Homology: How and Why
Hopf cyclic cohomology is a cohomological theory that aims
to provide some information about the cyclic cohomology of algebras
and coalgebras endowed with a symmetry from a Hopf algebra. This information
usually appears as certain subcomplex of the cyclic complex of the
algebras or coalgebras in question.
The theory initiated by Connes and Moscovici as a byproduct of their
study on the long standing computation of the local index formula
of the hypoelliptic operators. What they discovered is the miraculous
procedure that allowed them to detect the complicated local index
formula in the image of a very simple catachrestic map whose domain
was a new complex (Hopf cyclic complex) much simpler than the cyclic
complex of the groupoid action algebra.
Roughly speaking the roles of Lie groups and Lie algebras in differential
geometry are now given to Hopf algebras in noncommutative geometry.
Via this correspondence the Hopf cyclic cohomology replaces Lie algebra
cohomology and other cohomology theories associated to Lie groups.
The initial theory was modeled on Hopf algebras with a square identity
twisted antipode. Later on it covers Hopf algebras endowed with a
pair of character and grouplike so called modular pair in involution(MPI).
It was observed by HajacKhalkhaliRSommerh${\ddot{a}}$user that
the theory works perfectly for any Hopf algebras endowed with much
more general coefficients so called stable anti YetterDrinfeld
(SAYD) modules. These higher dimensional modulescomodules then
become essential in many occasions such as computations of Hopf cyclic
cohomology of Connes Moscovici type Hopf algebras, and also in the
definition of cup products in Hopf cyclic cohomology.
1. We start with the motivation and origin of the theory by recalling
two most important objects: characteristic classes of foliations and
index theory.
2. The second talk deals with the definition of the Hopf cyclic cohomology
for general Hopf algebras and general coefficients.
3. The next talk is devoted to the construction of those Hopf algebras
that are used (will be used) in the local index
formula of the corresponding Cartan geometries: General, Volume
Preserving, Symplectic, Contact, Projective. We call such Hopf
algebras the ConnesMoscovici type Hopf algebras.
4. In the fourth talk we develop a general theory by which one can
compute Hopf cyclic cohomology of Hopf algebras associated to a general
matched pair of Lie algebras or Lie groups.
5. Finally we explain various characteristic maps and cup products
that are cornerstones of Hopf cyclic cohomology. We apply these tools
to present, for the first time, the charactrestic classes of foliations,
in codimension higher than 1, as cyclic cocycle over the groupoid
action algebra.


