# HOPF-CYCLIC COHOMOLOGY

## A series of talks, Wednesdays 9:15, E46

Hopf-cyclic cohomology was recently discovered by Connes and Moscovici (e.g., see this paper) while studying the noncommutative geometry of foliations. It was subsequently linked with the Cuntz-Quillen formalism by Crainic, and with both algebraic and topological quantum groups as well as Hopf-algebraic crossed products (Akbarpour, Connes, Khalkhali, Kustermans, Moscovici, Murphy, Rognes, Tuset; cf. twisted cyclic cohomology). In July 2002, there was another breakthrough, due to Khalkhali and Rangipour. They managed to provide a general constructive framework for cyclic theories and significantly reduced one of the key computational proofs of Connes-Moscovici by providing conceptual arguments. In October 2002, the basic Khalkhali-Rangipour construction of equivariant and invariant Hopf-cyclic (co)homology was further generalised, and proofs were simplified. (See here for the most recent account.) Thus a rather general formalism that seems to essentialy tie up with some well-known Hopf-algebraic concepts, like dimodules and entwined categories, was obtained. The goal of this series of talks is to explain from scratch the basic constructions, then the aforementioned new results and open problems, and finally discuss some special cases and elementary but non-trivial examples coming from differential geometry, coquasitriangular Hopf algebras and locally compact quantum groups. Time permitting, we plan to append this series of talks by a lecture devoted to Hopf-algebroid versions of cyclic cohomology (see here and there). All interested are cordially invited to attend.

Wednesday 20.11.2002 09:15 E46

THE CATEGORICAL DEFINITION OF CYCLIC HOMOLOGY AND COHOMOLOGY

This is an introductory talk explaining general definitions such as: simplicial and cyclic objects, Hochschild (co)homology for pre-simplicial objects and cyclic (co)homology for pre-cyclic objects. These general definitions will be followed by standard examples of the Hochschild homology of algebras with coefficients in a bimodule, the cyclic homology of algebras, the Hochschild cohomology of coalgebras with coefficients in a bicomodule and the cyclic cohomology of coalgebras. Basic literature: Jean-Louis Loday, Cyclic Homology.

BOJANA FEMIC

Wednesday 27.11.2002 09:15 E46

STANDARD EXAMPLES OF HOCHSCHILD AND CYCLIC (CO)HOMOLOGY

First, it will be explained how the cyclic cohomology was originally defined and why this simple definition works in any Abelian category. Then the Hochschild homology of algebras with coefficients in a bimodule, the cyclic homology of algebras, the Hochschild cohomology of coalgebras with coefficients in a bicomodule and the cyclic cohomology of coalgebras will be discussed in detail. Recall that for the standard example of cyclic cohomology of algebras over a field containing rational numbers, both definitions coincide by a theorem of Loday-Quillen. The original definition was discovered by Alain Connes (independently by Boris Tsygan) as a receptacle of the Chern characer from K-homology, and allowed him to obtain the noncommutative index formula (see A. Connes, Noncommutative Differential Geometry, Inst. Hautes 'Etudes Sci. Publ. Math., vol.62, 257-360, 1985.) On the other hand, the already presented complicated-looking definition of cyclic cohomology works nicely in the characteristic-free framework and for non-unital algebras. It is also very useful in relating the Hochschild and cyclic cohomology. Basic literature: Jean-Louis Loday, Cyclic Homology.

BOJANA FEMIC

Wednesday 04.12.2002 09:15 E46

CYCLIC HOMOLOGY AND COHOMOLOGY OF HOPF SYSTEMS

A Hopf system over a Hopf algebra H is a triple (A,C,M), where A is a right H-comodule algebra, C is a right H-module coalgebra, and M is an involutive left H-dimodule. (All these notions will be explained.) The main goal is to show that natural invariant complexes of Hopf systems form cyclic modules, i.e., there exist faces, cyclic operators and degeneracies satisfying the axioms of the cyclic category. Thus invariant complexes of Hopf systems provide a unifying model for different types of cyclic homologies and cohomologies. This construction parallels to some extent the fundamental result of differential geometry that the de Rham cohomology ring of a manifold is isomorphic to the cohomology ring of differential forms invariant under the action of a compact group (Theorem 2.3 in Cohomology theory of Lie groups and Lie algebras", Trans. Amer. Math. Soc. vol.63, 1948, 85-124, by C. Chevalley and S. Eilenberg). The proof of the existence of faces and degeneracies (simplicial structure) will be provided in this talk. Due to its length, the key part, which is a proof of the existence of cyclic operators, will be postponed to the next session. Basic reference for this and the coming talk is a preliminary version of Hopf-cyclic homology and cohomology" by M. Khalkhali, B. Rangipour and PMH. It is available here.

PIOTR M. HAJAC

Wednesday 11.12.2002 09:15 E46

CYCLIC OPERATORS FOR INVARIANT HOPF SYSTEMS

The aim of this talk is to construct cyclic operators on invariant homology and cohomology complexes given by Hopf systems. The main point is to prove that the natural paracyclic operators that can be defined on equivariant complexes are still well defined on the invariant complexes. This completes the proof as the key identity $\tau_n^{n+1}=id$ is evidently satisfied on the invariant complexes. Time permitting, we will discuss some links between Hopf systems and Galois type extensions of noncommutative algebras and entwining structures.

PIOTR M. HAJAC

Wednesday 18.12.2002 09:15 E46

HOPF-CYCLIC (CO)HOMOLOGY WITH COEFFICIENTS IN STABLE ANTI-YETTER-DRINFEL'D MODULES

A beautiful new insight into the Hopf-cyclic cohomology was provided last Thursday by Yorck Sommerhaeuser. It allows one to replace involutive dimodules by anti-Yetter-Drinfel'd modules. As modular pairs in involution can always turn Yetter-Drinfel'd modules into anti-Yetter-Drinfel'd modules, the latter also appear in abundance. They generalise the involutive dimodules in a way that does not require any significant modification of the relevant proofs. The goal of this talk is to provide a proof of the well-definedness of the cyclic operators for the invariant complexes with coefficients in stable anti-Yetter-Drinfel'd modules, and to explore some immediate consequences of this new Yorck's idea. In particular, we will discuss in detail the classical Hopf fibration as a natural example of a stable (anti) Yetter-Drinfel'd module.

PIOTR M. HAJAC

Wednesday 08.01.2003 09:15 E46

STABLE ANTI-YETTER-DRINFEL'D MODULES AND SPECIAL CASES OF HOPF-CYCLIC THEORY

To begin with, we will show that the tensor product of a Yetter-Drinfel'd module with an anti Yetter-Drinfel'd module is again an anti Yetter-Drinfel'd module. Since modular pairs in involution are nothing but one-dimensional anti Yetter-Drinfel'd modules, such tensoring is a potential rich source of higher-dimensional examples of coefficients for Hopf-cyclic (co)homology. Next, we will show how an embedding of a homogeneous space in the group acting on it can be an example of a stable (anti) Yetter-Drinfel'd module. We will work out in detail the case of 2-sphere embedded in SU(2), and prove a lemma saying how one can turn quantum homogeneous spaces into stable anti Yetter-Drinfel'd modules. The second part of the talk is concerned with special cases: cyclic homology and cohomology of algebras as Hopf-cyclic theory for the trivial Hopf algebra and coefficients, and Connes-Moscovici cyclic cohomology of Hopf algebras as Hopf-cyclic theory for one-dimensional but non-trivial coefficients. Finally, we will discuss the simple relationship between the twisted cyclic cohomology of Kustermans, Murphy and Tuset, and the Hopf-cyclic cohomology of algebras with one-dimensional non-trivial coefficients.

PIOTR M. HAJAC

Wednesday 15.01.2003 09:15 E46

CONNES-MOSCOVICI CYCLIC COHOMOLOGY OF HOPF ALGEBRAS

This is an introduction to the subsequent talks concerning modular pairs in involution $(\delta,\sigma)$. It will be shown that the Connes-Moscovici cyclic cohomology of Hopf algebras is Hopf-cyclic theory for a Hopf algebra viewed as a right module coalgebra over itself and for a one dimensional but non-trivial anti-Yetter-Drinfel'd module. Then we will discuss $\delta$-invariant $\sigma$-traces as a way to map the cyclic cohomology of a Hopf algebra to the standard cyclic cohomology of an algebra on which this Hopf algebra acts.

PIOTR M. HAJAC

THE MODULAR PAIR IN INVOLUTION FOR CODIMENSION 1 FOLIATIONS

The Connes-Moscovici formalism is designed to handle foliations of any codimension. We focus on codimension 1 for the sake of simplicity. After reviewing some basic generalities and examples of foliations, we will define an action of certain vector fields an a certain cross-product algebra. One of these vector fields no longer acts as a derivation but its modified Leibniz rule still can be encoded in the coproduct of the appropriate modification of the universal enveloping Hopf algebra. This Hopf algebra is the Connes-Moscovici Hopf algebra for codimension 1 foliations. We end the talks by finding its unique modular pair in involution.

GASTON GARCIA

Wednesday 22.01.2003 09:15 E46

THE MODULAR PAIR IN INVOLUTION FOR CODIMENSION 1 FOLIATIONS (PART 2)

The Connes-Moscovici formalism is designed to handle foliations of any codimension. We focus on codimension 1 for the sake of simplicity. After reviewing some basic generalities and examples of foliations, we will define an action of certain vector fields an a certain cross-product algebra. One of these vector fields no longer acts as a derivation but its modified Leibniz rule still can be encoded in the coproduct of the appropriate modification of the universal enveloping Hopf algebra. This Hopf algebra is the Connes-Moscovici Hopf algebra for codimension 1 foliations. We end the talks by finding its unique modular pair in involution.

GASTON GARCIA

Wednesday 29.01.2003 09:15 E46

MODULAR PAIRS IN INVOLUTION FOR (CO)QUASITRIANGULAR HOPF ALGEBRAS

Hopf algebras with a modular pair in involution are quite natural structures. Interesting examples of Hopf algebras are (co)quasitriangular ones, and these come equipped with a modular pair in involution. Quasitriangular Hopf algebras have, in particular, a coalgebra structure that is almost cocommutative. Similarly their duals: coquasitriangular Hopf algebras (polynomial functions on quantum groups) are almost commutative algebras. Here the noncommutativity is controlled by a matrix R. Examples include SO_q(N), SU_q(N), and Sp_q(N) quantum groups. We introduce quasitriangular Hopf algebras, study their properties and show how each quasitriangular Hopf algebra canonically extends to a ribbon Hopf algebra. Given a ribbon Hopf algebra we immediately have a modular pair in involution. Similarly we study modular pairs in involution for coribbon algebras.

PAOLO ASCHIERI

Wednesday 05.02.2003 09:15 E46

EQUIVARIANT CYCLIC HOMOLOGY, YETTER-DRINFEL'D STRUCTURES AND QUANTUM GROUPS

In this talk we give a survey of equivariant cyclic homology with emphasis on some aspects related to Hopf algebras and quantum groups. In the first part we treat the basic case where G is a discrete group and explain the definition of the bivariant periodic theory HP^G_*(A,B) in this context. We will include some motivation and background material. From a technical point of view, the definition is a modification of the Cuntz-Quillen approach to ordinary cyclic homology. A completely new feature of the equivariant theory is that the fundamental objects are no longer complexes in the sense of homological algebra. We conclude the discussion with some results concerning computations of HP^G_*. In the second part we explain how the definitions can be modified in order to treat Hopf algebra actions. Here Yetter-Drinfel'd-type structures occur in a natural way. We study in particular the case of compact quantum groups. In this situation our construction leads to a natural definition of characteristic classes. We outline possible applications to Hopf-Galois extensions.

CHRISTIAN VOIGT
(Universitaet Muenster)