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Hopf-type cyclic cohomology via the Karboubi operator. (English) Zbl 1061.58005

Hajac, Piotr M. (ed.) et al., Noncommutative geometry and quantum groups. Proceedings of the Banach Center school/conference, Warsaw, Poland, September 17–29, 2001. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 61, 199-217 (2003).
From the introduction: In papers of A. Connes and H. Moscovici the explicit structure of a cocyclic module defining the so-called Hopf-type cyclic cohomology of a Hopf algebra was provided. Later on, M. Crainic showed that this cocyclic module could be obtained as the space of coinvariants of a Hopf-algebra action on some other cyclic module. Some further generalizations and developments were made by other autors (see the paper).
The purpose of this paper is to describe the Hopf-type cohomology of a Hopf algebra in terms of a subcomodule of the so-called algebra of non-commutative differential forms associated with the Hopf algebra. It turns out that to any modular pair in involution \((\delta, \sigma)\) one can associate a subcomplex of this differential algebra that is stable under the Karoubi operator \(\kappa\) or its twisted version \(\kappa_\xi\).
In addition to giving a new point of view on this cohomology theory, this approach seems to have some virtues of its own. For instance, one can try to define some similar sort of cyclic homology when a more general object is used instead of a modular pair. Besides this, it can be used to establish bridges between cyclic cohomology and Hopf-Galois theory.
For the entire collection see [Zbl 1024.00070].

MSC:

58B34 Noncommutative geometry (à la Connes)
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