Cup products in Hopf-cyclic cohomology

Khalkhali, Masoud; Rangipour, Bahram

doi:10.1016/j.crma.2004.10.025

Cited by 27 publications

(60 citation statements)

References 19 publications

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“…This is done in the homotopy category of towers of super complexes which is homotopy equivalent to the derived category of mixed complexes by Quillen [21]. In their setup Gorokhovsky [10], and later Khalkhali and Rangipour [18] also used mixed complexes to obtain their cohomology classes, and (H-invariant) closed graded (co)traces to implement their pairings. This is akin to Connes' use of closed graded traces to implement the ordinary cup product in cyclic cohomology [3, III.1, Thm.…”

Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

“…I would like to thank both institutions for their generous support and hospitality. I thank Antoine Touze for his help on Lemma 3.9, and Bahram Rangipour for explaining few key points in [18]. Last but not the least, I would like to thank Henri Moscovici for the discussions we had about this work, and many other things, over our daily coffee breaks.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 6.4. Let A be a H-module algebra and C be a H-module coalgebra acting on A equivariantly over H. The pairings defined in [4,10,7,18,15,23,22] are naturally isomorphic as natural transformations of isomorphic double functors.…”

Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

“…The pairing comes from Proposition 4.5 followed by Lemma 6.1. Our aim is to show that pairings defined in [4,10,7,18,15,23,22] in Hopf-cyclic cohomology are naturally isomorphic as natural transformations of double functors. There are certain variations between these pairings: The Connes-Moscovici, Gorokhovsky and Crainic pairings are defined for C = H, q = 0 and only for M = k σ,δ the canonical 1-dimensional SAYD module associated with a modular pair involution in H. The Rangipour-Khalkhali pairing, and the pairing Rangipour defined in [22], are defined for an arbitrary module coalgebra C acting equivariantly on a module algebra A, and for arbitrary bidegree (p, q) with an arbitrary SAYD module as coefficients [12].…”

Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

“…There are numerous such products and pairings in the literature [10,7,18,15,23,22] which extend the characteristic map defined by Connes and Moscovici [4]. Let H be a Hopf algebra, A be a H-module algebra and M be an arbitrary H-module/comodule.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Uniqueness of pairings in Hopf-cyclic cohomology

Kaygun

2010

J K-Theor

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Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

Section: Pairings In Cyclic (Co)homologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Uniqueness of pairings in Hopf-cyclic cohomology

Kaygun

2010

J K-Theor

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On the Hopf-type Cyclic Cohomology with Coefficients

Nikonov

Sharygin²

Trends in Mathematics

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Transgressions of the Godbillon-Vey Class and Rademacher functions

Connes

Moscovici

Noncommutative Geometry and Number Theory

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we used an ad hoc pairing with modular symbols to convert the Godbillon-Vey class [δ 1 ] ∈ HC 1 (H 1 ) into the Euler class of GL + (2, Q).In this paper we provide a completely conceptual explanation for the above pairing and at the same time extend it to the higher weight case. This is achieved by constructing out of modular symbols H 1 -invariant 1-traces that support characteristic maps for certain actions of H 1 on A Q , canonically associated to modular forms. Moreover, we show that the image of the Godbillon-Vey class through these characteristic homomorphisms, obtained by the cup product between [δ 1 ] and the invariant 1-traces, transgresses to secondary data. For the projective action determined by the Ramanujan connection the transgression takes place within the Euler class, in a manner that ressembles the K-homological transgression in the context of SU q (2) [5], and leads to the classical Rademacher function [23]. For the actions associated to cusp forms of higher weight the transgressed classes implement the Eichler-Shimura isomorphism. The actions corresponding to Eisenstein series give rise by transgression to higher Dedekind sums and generalized Rademacher functions ([28, 20]), or equivalently to the Eisenstein cocycle of [28].Generalized Dedekind sums have been related to special values of L-functions in the work of C. Meyer on the class-number formula [18,19], and higher Dedekind sums appear in the work of Siegel [25,26] and Zagier [29] on the values at non-positive integers of partial zeta functions over real quadratic fields. Eisenstein cocycles were employed by Stevens [28] and by Sczech [24] in order to compute these values more efficiently. The fact that these notions can be interpreted as secondary invariants is reminiscent of the secondary nature of the regulator invariants (cf. e.g. [10]) that are involved in the expression of the special values at non-critical points of L-functions associated to number fields (see e.g. [16], [30]).

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Cup products in Hopf-cyclic cohomology

Cited by 27 publications

References 19 publications

Uniqueness of pairings in Hopf-cyclic cohomology

Uniqueness of pairings in Hopf-cyclic cohomology

On the Hopf-type Cyclic Cohomology with Coefficients

Transgressions of the Godbillon-Vey Class and Rademacher functions

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