Bialgebra cohomology and exact sequences

Julien Bichon

doi:10.5802/crmath.659

Article de recherche - Algèbre

Bialgebra cohomology and exact sequences
[Cohomologie de bigèbres et suites exactes]

Julien Bichon ¹

¹ Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1475-1483.

Résumé
Abstract

Nous montrons comment les cohomologies de Gertstenhaber–Schack de deux algèbres de Hopf imbriquées dans une suite exacte courte sont reliées, quand le troisième facteur est cosemisimple de dimension finie. Nous en déduisons une preuve rapide du calcul de la cohomologie de bigèbre des algèbres de Hopf cosouveraines universelles dans le cas générique, établi récemment par Baraquin, Franz, Gerhold, Kula et Tobolski.

We show how the bialgebra cohomologies of two Hopf algebras involved in an exact sequence are related, when the third factor is finite-dimensional cosemisimple. As an application, we provide a short proof of the computation of the bialgebra cohomology of the universal cosovereign Hopf algebras in the generic (cosemisimple) case, done recently by Baraquin, Franz, Gerhold, Kula and Tobolski.

Reçu le : 2023-10-09
Révisé le : 2024-05-29
Accepté le : 2024-06-17
Publié le : 2024-11-14

Zbl

DOI : 10.5802/crmath.659

Classification : 16T05, 16E40
Keywords: Hopf algebras, bialgebra cohomology, Yetter–Drinfeld modules
Mots-clés : Algèbres de Hopf, cohomologie des bigèbres, modules de Yetter–Drinfeld

Affiliations des auteurs :

Julien Bichon ¹

¹ Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Julien Bichon},
     title = {Bialgebra cohomology and exact sequences},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1475--1483},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.659},
     zbl = {07945489},
     language = {en},
}

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Julien Bichon. Bialgebra cohomology and exact sequences. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1475-1483. doi : 10.5802/crmath.659. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.659/

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