THEMATIC PROGRAMS

We present a categorical approach offering a conceptual simplification of Hopf-cyclic 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 Hopf-cyclic theory based on algebras, coalgebras, Hopf bialgebroids and coefficients in stable anti-Yetter-Drinfeld modules. In particular, algebras are replaced by monoidal functors, stable anti-Yetter-Drinfeld 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 Bohm-Stefan or that of Kaledin.

Definition of a quantum groupoid in the C*-algebra framework

In this talk, I will report on my recent (and on-going) 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 non-degenerate, 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 on-going, 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.

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 K-groups in terms of certain weighted graphs, based on the categorical structure of such modules. Based on joint work with Kenny De Commer.

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 H-operation, 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 H-operations with connections.

Quantization of coadjoint orbits

We present an operator algebraic approach to the quantization of certain coadjoint orbits, with modified Poisson bracket, in the Poisson-Lie duals of semi-simple compact Lie groups.

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 Yetter-Drinfeld
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.

New remarks on the Dirac operators on quantised Hermitian symmetric spaces

In this joint work with Matthew Tucker-Simmons (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.

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.

On the characteristic classes of symplectic foliations via Hopf-cyclic cocycles

We report on our on-going 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 Connes-Moscovici 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 Hopf-algebra 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.

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 Byung-Jay Kahng (Canisius College Buffalo - USA).

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.

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 non-abelian 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 semi-continuous weights on ${C^*}$-algebras and normal weights on von Neumann algebras, in relation with various aspects of the Tomita-Takesaki 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.

Hopf-Cyclic 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 sub-complex 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 hypo-elliptic 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 group-like so called modular pair in involution(MPI). It was observed by Hajac-Khalkhali-R-Sommerh${\ddot{a}}$user that the theory works perfectly for any Hopf algebras endowed with much more general coefficients so called stable anti Yetter-Drinfeld (SAYD) modules. These higher dimensional modules-comodules 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 Connes-Moscovici 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.