On the Hopf-type Cyclic Cohomology with Coefficients

Pairings in Hopf cyclic cohomology of algebras and coalgebras with coefficients

“…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]. …”

“…Methods used in that paper are quite different from those of [9]. In the paper [12], the authors gave a brief outline of the recipe which allows one to extend the methods used by Crainic to obtain another construction of the pairing, similar to (12). We shall denote this pairing by ] 0 .…”

“…This question was addressed in the paper of M. Khalkhali and B. Rangipour [9] and independently in the paper of the present authors, [12]. In both papers a pairing was constructed in the sense described above.…”

“…HC pCq .A/ (12) was defined which extends to higher dimensions the Connes-Moscovici characteristic map (see [2], [1] and [7]) W HC H .C; M/ ! HC .A/, constructed for an equivariant trace W M˝A !…”

“…In both papers a pairing was constructed in the sense described above. While the construction of [9] was based on ideas of A. Connes, the paper [12] was inspired by methods of M. Crainic and gave a pairing between an appropriate version of the higher traces on A and the Hopf cyclic cohomology of a coalgebra. Below we shall refer to this pairing as a"Crainic-type pairing".…”