Toward 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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Detalles Bibliográficos
Autores: Ardizzoni, Alessandro, El Kaoutit, Laiachi, Saracco, Paolo
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2023
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/172814
Acceso en línea:https://hdl.handle.net/11441/172814
https://doi.org/10.5565/PUBLMAT6712301
Access Level:acceso abierto
Palabra clave:(co)commutative Hopf algebroids
affine groupoid schemes
differentiation and integration
K¨ahler module
Lie–Rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
finite dual
Tannaka reconstruction
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spelling Toward differentiation and integration between Hopf algebroids and Lie algebroidsArdizzoni, AlessandroEl Kaoutit, LaiachiSaracco, Paolo(co)commutative Hopf algebroidsaffine groupoid schemesdifferentiation and integrationK¨ahler 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.Universitat Autònoma de BarcelonaÁlgebra2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/172814https://doi.org/10.5565/PUBLMAT6712301reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésPublicacions Matemàtiques, 67 (1), 3-88.https://dx.doi.org/10.5565/PUBLMAT6712301info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1728142026-06-17T12:51:07Z
dc.title.none.fl_str_mv Toward differentiation and integration between Hopf algebroids and Lie algebroids
title Toward differentiation and integration between Hopf algebroids and Lie algebroids
spellingShingle Toward differentiation and integration between Hopf algebroids and Lie algebroids
Ardizzoni, Alessandro
(co)commutative Hopf algebroids
affine groupoid schemes
differentiation and integration
K¨ahler module
Lie–Rinehart algebras
Lie algebroids
Lie groupoids
Malgrange groupoids
finite dual
Tannaka reconstruction
title_short Toward differentiation and integration between Hopf algebroids and Lie algebroids
title_full Toward differentiation and integration between Hopf algebroids and Lie algebroids
title_fullStr Toward differentiation and integration between Hopf algebroids and Lie algebroids
title_full_unstemmed Toward differentiation and integration between Hopf algebroids and Lie algebroids
title_sort Toward differentiation and integration between Hopf algebroids and Lie algebroids
dc.creator.none.fl_str_mv Ardizzoni, Alessandro
El Kaoutit, Laiachi
Saracco, Paolo
author Ardizzoni, Alessandro
author_facet Ardizzoni, Alessandro
El Kaoutit, Laiachi
Saracco, Paolo
author_role author
author2 El Kaoutit, Laiachi
Saracco, Paolo
author2_role author
author
dc.contributor.none.fl_str_mv Álgebra
dc.subject.none.fl_str_mv (co)commutative Hopf algebroids
affine groupoid schemes
differentiation and integration
K¨ahler 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
K¨ahler 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 2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/172814
https://doi.org/10.5565/PUBLMAT6712301
url https://hdl.handle.net/11441/172814
https://doi.org/10.5565/PUBLMAT6712301
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Publicacions Matemàtiques, 67 (1), 3-88.
https://dx.doi.org/10.5565/PUBLMAT6712301
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Universitat Autònoma de Barcelona
publisher.none.fl_str_mv Universitat Autònoma de Barcelona
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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