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THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP
Jose N. Alonso ´Alvarez,Ramon Gonz´alez Rodr´ıguez,Jose M. Fern´andez Vilaboa 대한수학회 2017 대한수학회지 Vol.54 No.2
Let $H$ be a cocommutative faithfully flat Hopf quasigroup in a strict symmetricmonoidal category with equalizers. In this paper we introduce the notion of (strong) Galois $H$-object and we prove that the set of isomorphism classes of (strong) Galois $H$-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of $H$-Galois objects introduced by Chase and Sweedler.
THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP
Alvarez, Jose N. Alonso,Rodriguez, Ramon Gonzalez,Vilaboa, Jose M. Fernandez Korean Mathematical Society 2017 대한수학회지 Vol.54 No.2
Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.
Monoidal functors and exact sequences of groups for Hopf quasigroups
Jose N. Alonso Alvarez,Jose M. Fernandez Vilaboa,Ramon Gonzalez Rodriguez 대한수학회 2021 대한수학회지 Vol.58 No.2
In this paper we introduce the notion of strong Galois $H$-progenerator object for a finite cocommutative Hopf quasigroup $H$ in a symmetric monoidal category ${\sf C}$. We prove that the set of isomorphism classes of strong Galois $H$-progenerator objects is a subgroup of the group of strong Galois $H$-objects introduced in \cite{JKMS}. Moreover, we show that strong Galois $H$-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if $H$ is finite, we find exact sequences of Picard groups related with invertible left $H$-(quasi)modules and an isomorphism $Pic(_{{\sf H}}{\sf Mod})\cong Pic({\sf C})\oplus G(H^{\ast})$ where $Pic(_{{\sf H}}{\sf Mod})$ is the Picard group of the category of left $H$-modules, $Pic({\sf C})$ the Picard group of ${\sf C}$, and $G(H^{\ast})$ the group of group-like morphisms of the dual of $H$.