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Pairings in Hopf cyclic cohomology of algebras and coalgebras with coefficients

Published online by Cambridge University Press:  15 March 2010

I. Nikonov  and
G. Sharygin
I. Nikonov
Affiliation:
Dept. of Differential Geometry, Moscow State University, Leninskie Gory, Moscow, Russia, nikonov@mech.math.msu.su
G. Sharygin
Affiliation:
ITEP, ul. B. Cheremushkinskaja, 25, Moscow, Russia, sharygin@itep.ru
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Abstract

This paper is concerned with the theory of cup products in the Hopf cyclic cohomology of algebras and coalgebras. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving the noncommutative Weil algebra. Then we introduce the notion of higher -twisted traces and use a generalization of the Quillen and Crainic constructions (see [14] and [3]) to define the cup product. We discuss the relation of the cup product above and S-operations on cyclic cohomology. We show that the product we define can be realized as a combination of the composition product in bivariant cyclic cohomology and a map from the cyclic cohomology of coalgebras to bivariant cohomology. In the last section, we briefly discuss the relation of our constructions with that in [9]. More precisely, we propose still another construction of such pairings which can be regarded as an intermediate step between the “Crainic” pairing and that of [9]. We show that it coincides with what in [9] and as far its relation to Crainic's construction is concerned, we reduce the question to a discussuion of a certain map in cohomology (see the question at the end of section 5).

The results of the current paper were announced in [12].

Keywords

cyclic cohomology with coefficientsHopf algebras
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 2 , April 2010 , pp. 289 - 348
Copyright
Copyright © ISOPP 2010

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References

1.Connes, A., Moscovici, H.: Cyclic cohomology and Hopf algebras, Lett. Math. Phys. 52, 97–108 (2000)CrossRefGoogle Scholar
2.Connes, A., Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198, 199–246 (1998)CrossRefGoogle Scholar
3.Crainic, M.: Cyclic cohomology of Hopf algebras, J. Pure Appl. Algebra 166, 29–66 (2002)CrossRefGoogle Scholar
4.Cuntz, J., Quillen, D.: Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8(2), 251–289 (1995)CrossRefGoogle Scholar
5.Cuntz, J., Quillen, D.: Cyclic homology and nonsingularity, J. Amer. Math. Soc. 8(2), 373–442 (1995)CrossRefGoogle Scholar
6.Goodwillie, T.G.: Cyclic homology, derivations, and the free loopspace, Topology 24(2), 187–215 (1985)CrossRefGoogle Scholar
7.Hajac, P.M., Khalkali, M., Rangipour, B., Sommerhäuser, M.: Hopf cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338(9), 667–672 (2004) (also available as preprint arXiv:math.KT/0306288 v.2)CrossRefGoogle Scholar
8.Karoubi, M.: Cohomologie cyclique et K-théorie, Astérisque 149 (1987)Google Scholar
9.Khalkhali, M., Rangipour, B.: Cup Products in Hopf cyclic Cohomology, C. R. Math. Acad. Sci. Paris 340, 9–14 (2005)CrossRefGoogle Scholar
10.Khalkhali, M., Rangipour, B.: A new cyclic module for Hopf algebras, K-theory 27, 111–131 (2002)CrossRefGoogle Scholar
11.Loday, J.L.: Cyclic homology, Springer-Verlag (1998)CrossRefGoogle Scholar
12.Nikonov, I., Sharygin, G.: On the Hopf-type cyclic cohomology with coefficients. In: C*- algebras and Elliptic Theory. Trends in Mathematics, pp.203–212. Birkhäuser Verlag, Basel, (2006)Google Scholar
13.Quillen, D.: Algebra cochains and cyclic cohomology, Publ. Math. IHES 68, 139–174 (1989)CrossRefGoogle Scholar
14.Quillen, D.: Chern-Simons forms and cyclic cohomology. In: The interface of Mathematics and particle Physics, 117–134, Oxford, 1988Google Scholar
15.Sharygin, G.: A new construction of characteristic classes for noncommutative algebraic principal bundles, Banach Center Publ. 61, 219–230 (2003)CrossRefGoogle Scholar
16.Taileffer, R.: Cyclic homology of Hopf algebras, K-Theory 24, 69–85 (2001)CrossRefGoogle Scholar