Towards differentiation and integration between Hopf algebroids and Lie algebroids

In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base a...

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Autores: Ardizzoni, Alessandro|||0000-0001-7384-611X, El Kaoutit, Laiachi|||0000-0002-8782-8219, Saracco, Paolo|||0000-0001-5693-7722
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:271779
Acceso en línea:https://ddd.uab.cat/record/271779
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712301
Access Level:acceso abierto
Palabra clave:(co)commutative hopf algebroids
Affine groupoid schemes
Differentiation and integration
Kähler module
Lie-rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
Finite dual
Tannaka reconstruction
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spelling Towards differentiation and integration between Hopf algebroids and Lie algebroidsArdizzoni, Alessandro|||0000-0001-7384-611XEl Kaoutit, Laiachi|||0000-0002-8782-8219Saracco, Paolo|||0000-0001-5693-7722(co)commutative hopf algebroidsAffine groupoid schemesDifferentiation and integrationKähler moduleLie-rinehart algebrasLie algebroidsLie groupoidsMalgrange groupoidsFinite dualTannaka reconstructionIn this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors from the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented. 22023-01-0120232023-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/271779https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712301reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77033-Popen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2717792026-06-06T12:50:31Z
dc.title.none.fl_str_mv Towards differentiation and integration between Hopf algebroids and Lie algebroids
title Towards differentiation and integration between Hopf algebroids and Lie algebroids
spellingShingle Towards differentiation and integration between Hopf algebroids and Lie algebroids
Ardizzoni, Alessandro|||0000-0001-7384-611X
(co)commutative hopf algebroids
Affine groupoid schemes
Differentiation and integration
Kähler module
Lie-rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
Finite dual
Tannaka reconstruction
title_short Towards differentiation and integration between Hopf algebroids and Lie algebroids
title_full Towards differentiation and integration between Hopf algebroids and Lie algebroids
title_fullStr Towards differentiation and integration between Hopf algebroids and Lie algebroids
title_full_unstemmed Towards differentiation and integration between Hopf algebroids and Lie algebroids
title_sort Towards differentiation and integration between Hopf algebroids and Lie algebroids
dc.creator.none.fl_str_mv Ardizzoni, Alessandro|||0000-0001-7384-611X
El Kaoutit, Laiachi|||0000-0002-8782-8219
Saracco, Paolo|||0000-0001-5693-7722
author Ardizzoni, Alessandro|||0000-0001-7384-611X
author_facet Ardizzoni, Alessandro|||0000-0001-7384-611X
El Kaoutit, Laiachi|||0000-0002-8782-8219
Saracco, Paolo|||0000-0001-5693-7722
author_role author
author2 El Kaoutit, Laiachi|||0000-0002-8782-8219
Saracco, Paolo|||0000-0001-5693-7722
author2_role author
author
dc.subject.none.fl_str_mv (co)commutative hopf algebroids
Affine groupoid schemes
Differentiation and integration
Kähler module
Lie-rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
Finite dual
Tannaka reconstruction
topic (co)commutative hopf algebroids
Affine groupoid schemes
Differentiation and integration
Kähler module
Lie-rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
Finite dual
Tannaka reconstruction
description In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors from the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented.
publishDate 2023
dc.date.none.fl_str_mv 2
2023-01-01
2023
2023-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/271779
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712301
url https://ddd.uab.cat/record/271779
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6712301
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2016-77033-P
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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