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Cyclic cohomology of (extended) Hopf algebras. (English) Zbl 1100.19001

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, 59-89 (2003).
The authors review recent developments on the cohomology of Hopf algebras, extending the initial work of A. Connes and H. Moscovici [e.g., Commun. Math. Phys. 198, 199–246 (1998; Zbl 0940.58005), Lett. Math. Phys. 48, 97–108 (1999; Zbl 0941.16024)] via work of M. Crainic [J. Pure Appl. Algebra 166, 29–66 (2002; Zbl 0999.19004)], the authors [M. Khalkhali and B. Rangipour, K-Theory 28, 183–205 (2003; Zbl 1028.58006), Lett. Math. Phys. 70, 259–272 (2004; Zbl 1067.58007)], and the authors together with P. M. Hajac and Y. Sommerhäuser [C. R., Math., Acad. Sci. Paris 338, 667–672 (2004; Zbl 1064.16006)]. Following preliminary sections on Hopf algebras and cyclic modules (in the sense of Connes), there are extensive sections on the cyclic cohomology of Hopf algebras and of extended Hopf algebras. (The latter are closely related to, but different from, Hopf algebroids; they were introduced by the authors [op. cit.] in order to be able to define cocyclic modules for appropriate systems.) The paper concludes with briefer sections reviewing cohomology of smash products and Hopf-cyclic homology with coefficients.
For the entire collection see [Zbl 1024.00070].

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
18G60 Other (co)homology theories (MSC2010)
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