Hopf-cyclic homology and cohomology with coefficients

Piotr M. Hajac; Masoud Khalkhali; Bahram Rangipour; Yorck Sommerhäuser

doi:10.1016/j.crma.2003.11.036

Homological Algebra/Topology

Hopf-cyclic homology and cohomology with coefficients

Presented by: Alain Connes

Piotr M. Hajac ^1,²; Masoud Khalkhali ³; Bahram Rangipour ³; Yorck Sommerhäuser ⁴

¹ Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa, 00-956 Poland
² Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa, 00-682 Poland
³ Department of Mathematics, University of Western Ontario, London ON, Canada
⁴ Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany

Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 667-672.

Abstract
Résumé

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra.

Suivant l'idée d'un complexe différentiel invariant, nous construisons des modules cycliques de type général qui fournissent un dénominateur commun aux théories cycliques connues. Le caractère cyclique de ces modules est gouverné par des structures Hopf-algébriques. Nous montrons que l'existence d'un opérateur cyclique oblige à une modification de la condition de compatibilité de Yetter–Drinfeld et mène au concept de module anti-Yetter–Drinfeld stable. Ce module joue le rôle d'espace de coefficients pour la cohomologie de modules algèbres et de modules cogèbres ainsi obtenue, ainsi que pour l'homologie et la cohomologie cycliques de comodules algèbres. Comme l'ont fait Connes et Moscovici pour leur cohomologie, nous montrons qu'il existe un appariement entre la cohomologie cyclique d'un module cogèbre agissant sur un module algèbre et les 0-cycles fermés sur ce dernier. Cet appariement prend ses valeurs dans la cohomologie cyclique usuelle de l'algèbre. De façon similaire, nous établissons un appariement analogue entre les 0-cycles fermés d'un module cogèbre et la cohomologie cyclique d'un module algèbre.

Received: 2003-07-31
Accepted: 2003-11-12
Published online: 2004-03-27

DOI: 10.1016/j.crma.2003.11.036

Author's affiliations:

Piotr M. Hajac ^{1, 2}; Masoud Khalkhali ³; Bahram Rangipour ³; Yorck Sommerhäuser ⁴

¹ Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa, 00-956 Poland
² Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa, 00-682 Poland
³ Department of Mathematics, University of Western Ontario, London ON, Canada
⁴ Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany

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     title = {Hopf-cyclic homology and cohomology with coefficients},
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Piotr M. Hajac; Masoud Khalkhali; Bahram Rangipour; Yorck Sommerhäuser. Hopf-cyclic homology and cohomology with coefficients. Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 667-672. doi : 10.1016/j.crma.2003.11.036. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.11.036/

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