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Theses

Théories homologiques des algèbres de Hopf

Abstract : In this thesis, we study homological and cohomological theories adapted to Hopf algebras.
In the first part, we unify various cohomological theories for Hopf algebras. Two of them were introduced by M. Gerstenhaber and S.D. Schack; one is without coefficients and is related to the cohomology adapted to the study of deformations of Hopf algebras, the other is a theory with coefficients (they are Hopf bimodules). The third is a generalization of the cohomology which was defined by C. Ospel, it is also a theory with coefficients. To unify these theories, we identify them with the Ext functor on an associative algebra defined by C. Cibils and M. Rosso which is an `enveloping algebra' associated to the Hopf algebra. Next, we establish explicit formulae for a cup-product on two of these cohomologies, and prove that this product corresponds to the Yoneda product of extensions.We also prove Morita invariance for these cohomologies.
The second part of the thesis is devoted to the study of a cyclic homology for Hopf algebras. It is dual to the cyclic cohomology introduced by A. Connes and H. Moscovici. We study some properties, then consider the case of group algebras. We interpret some decompositions (those of Burghelea and Karoubi-Villamayor) of the classical cyclic homology of group algebras in terms of Connes and Moscovici's cyclic homology of Hopf algebras. Then, we establish a decomposition formula (similar to that of Karoubi-Villamayor) for the cyclic homology of a cocommutative Hopf algebra (which generalizes a result of Khalkhali and Rangipour).
Finally, we compute some examples of homologies: the classical cyclic homology of truncated quiver algebras, as well as Connes and Moscovici's cyclic homology in the special case of the Taft algebras, and the Hochschild and classical cyclic homologies of the Auslander algebras of the Taft algebras.
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https://tel.archives-ouvertes.fr/tel-00001150
Contributor : Rachel Taillefer <>
Submitted on : Tuesday, February 26, 2002 - 6:12:54 PM
Last modification on : Thursday, January 11, 2018 - 6:15:40 AM
Long-term archiving on: : Tuesday, September 7, 2010 - 4:31:08 PM

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  • HAL Id : tel-00001150, version 1

Citation

Rachel Taillefer. Théories homologiques des algèbres de Hopf. Mathématiques [math]. Université Montpellier II - Sciences et Techniques du Languedoc, 2001. Français. ⟨tel-00001150⟩

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