Strong Hopf modules for weak Hopf quasigroups

J. N. Alonso Álvarez¹ ; J. M. Fernández Vilaboa² ; R. González Rodríguez³

¹ Departamento de Matemáticas Universidad de Vigo Campus Universitario Lagoas-Marcosende E-36280 Vigo, Spain
² Departamento de Álxebra Universidad de Santiago de Compostela E-15771 Santiago de Compostela, Spain
³ Departamento de Matemática Aplicada II Universidad de Vigo Campus Universitario Lagoas-Marcosende E-36310 Vigo, Spain

Colloquium Mathematicum, Tome 148 (2017) no. 2, pp. 231-246.

Résumé

This paper is a further step in the study of the theory of modules associated to a weak Hopf quasigroup $H$. We introduce the category of strong $H$-Hopf modules, and we prove that there exists an adjoint equivalence between this category and the category of right modules over the image of the target morphism of $H$. In the Hopf quasigroup setting every Hopf module is strong, and we recover the results of Brzeziński. Also, in the weak Hopf case, every Hopf module is strong, and we generalize the theorem proved by Böhm, Nill and Szlachányi that contains as a particular instance the categorical equivalence associated to the category of Hopf modules for a Hopf algebra $H$.

Détail

DOI : 10.4064/cm6967-6-2016

Keywords: paper further step study theory modules associated weak hopf quasigroup introduce category strong h hopf modules prove there exists adjoint equivalence between category category right modules image target morphism hopf quasigroup setting every hopf module strong recover results brzezi ski weak hopf every hopf module strong generalize theorem proved nill szlach nyi contains particular instance categorical equivalence associated category hopf modules hopf algebra nbsp