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)