### Abstract

In the recent years there appeared a number of papers, generalising the Connes-Moscovici's construction of cyclic cohomology of Hopf algebras. It turned out that in this case it is possible to introduce certain coefficients module in the cyclic cohomology (see, e.g. [1]). In the present talk (poster presentation) we investigate further this construction: we show how the X-complex formalism of Cuntz and Quillen (see [2]) can be extended to embrace the coefficients, and use this result to introduce the bivariant cyclic theory with coefficients. It turns out that the composition product in bivariant theory can be used to introduce the pairing between cohomology of an algebra and a coalgebra. We show, that this construction is a generalisation of Crainic's pairing ([3]) and that under certain conditions on the coalgebra it coincides with the pairing, introduced by Khalkhali and Rangipour ([4]). This talk is partly based on the paper [5] of Igor Nikonov and G.Sh..

[1] P.M.Hajac, M.Khalkali, B.Rangipour, M.Sommerhaeuser: Hopf-cyclic homology and cohomology with coefficients; C. R. Math. Acad. Sci. Paris 338, (2004), no. 9, 667-672 (also available as preprint arXiv:math.KT/0306288 v.2) [2] J.Cuntz, D.Quillen: Cyclic homology and nonsingularity; J. Amer. Math. Soc., 8, n.2 (1995) 373-442 [3] M.Crainic: Cyclic cohomology of Hopf algebras; J. Pure Appl. Algebra, 166, (2002) 29-66 [4] M.Khalkhali, B.Rangipour: Cup Products in Hopf-Cyclic Cohomology; available as preprint at arXiv:math.QA/0411003 v1 [5] I.Nikonov, G.Sharygin: On the Hopf-type Cyclic Cohomology with Coeﬃcients, C* -algebras and Elliptic Theory Trends in Mathematics, 203–212, 2006 Birkhaeuser Verlag Basel/Switzerland