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Correspondence theorems for Hopf algebroids with applications to affine groupoids

Part of: Hopf algebras, quantum groups and related topics Special categories

Published online by Cambridge University Press:  11 April 2023

Laiachi El Kaoutit ,
Aryan Ghobadi ,
Paolo Saracco Open the ORCID record for Paolo Saracco [Opens in a new window]  and
Joost Vercruysse Open the ORCID record for Joost Vercruysse [Opens in a new window]
Laiachi El Kaoutit
Affiliation:
Facultad de Ciencias, Departamento de Álgebra and IMAG, Universidad de Granada, Fuente Nueva s/n. E810071 Granada, Spain e-mail: kaoutit@ugr.es
Aryan Ghobadi
Affiliation:
School of Mathematics, Queen Mary University of London, Mile End Road, E1 4NS London, UK e-mail: a.ghobadi@qmul.ac.uk
Paolo Saracco
Affiliation:
Département de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Brussels, Belgium e-mail: paolo.saracco@ulb.be
Joost Vercruysse*
Affiliation:
Département de Mathématique, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Brussels, Belgium e-mail: paolo.saracco@ulb.be
*
e-mail: joost.vercruysse@ulb.be
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Abstract

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

Keywords

Hopf algebroidsnormal idealsaffine groupoidsnormal affine subgroupoidscomodule algebrasfinite groupoidsHopf algebroids of functionsGalois correspondence

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 16T15: Coalgebras and comodules; corings 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories)
Type
Article
Information
Canadian Journal of Mathematics , Volume 76 , Issue 3 , June 2024 , pp. 830 - 880
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Paolo Saracco is a Chargé de Recherches of the Fonds de la Recherche Scientifique—FNRS and a member of the “National Group for Algebraic and Geometric Structures and their Applications” (GNSAGA-INdAM). Aryan Ghobadi is a postdoctoral researcher under the EPSRC grant EP/W522508/1 and would also like to thank the LMS for the travel grant ECR-1920-42, which allowed the author to be included in this project. Joost Vercruysse would like to thank the Fédération Wallonie-Bruxelles (FWB) for support through the ARC project “From algebra to combinatorics, and back.”

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