Simply transitive NIL-affine actions of solvable Lie groups

Autores
Deré, Jonas; Origlia, Marcos Miguel
Año de publicación
2021
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Every simply connected and connected solvable Lie group G admits a simply transitive action on a nilpotent Lie group H via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups G can act simply transitively on which Lie groups H. So far, the focus was mainly on the case where G is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action : G Aff(H) is simply transitive by looking only at the induced morphism p: G → aff(h) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group G acts simply transitively on a given nilpotent Lie group H, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull,whichwe also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.
Fil: Deré, Jonas. Katholikie Universiteit Leuven; Bélgica
Fil: Origlia, Marcos Miguel. Monash University; Australia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Materia
ALGEBRAIC HULL
LIE ALGEBRA
SEMISIMPLE SPLITTING
SIMPLY TRANSITIVE ACTION
SOLVABLE LIE GROUP
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/172726

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network_name_str CONICET Digital (CONICET)
spelling Simply transitive NIL-affine actions of solvable Lie groupsDeré, JonasOriglia, Marcos MiguelALGEBRAIC HULLLIE ALGEBRASEMISIMPLE SPLITTINGSIMPLY TRANSITIVE ACTIONSOLVABLE LIE GROUPhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Every simply connected and connected solvable Lie group G admits a simply transitive action on a nilpotent Lie group H via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups G can act simply transitively on which Lie groups H. So far, the focus was mainly on the case where G is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action : G Aff(H) is simply transitive by looking only at the induced morphism p: G → aff(h) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group G acts simply transitively on a given nilpotent Lie group H, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull,whichwe also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.Fil: Deré, Jonas. Katholikie Universiteit Leuven; BélgicaFil: Origlia, Marcos Miguel. Monash University; Australia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaDe Gruyter2021-09-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/172726Deré, Jonas; Origlia, Marcos Miguel; Simply transitive NIL-affine actions of solvable Lie groups; De Gruyter; Forum Mathematicum; 33; 5; 1-9-2021; 1349-13670933-77411435-5337CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1515/forum-2020-0114info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/document/doi/10.1515/forum-2020-0114/htmlinfo:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2004.12774info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:21:01Zoai:ri.conicet.gov.ar:11336/172726instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:21:01.455CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Simply transitive NIL-affine actions of solvable Lie groups
title Simply transitive NIL-affine actions of solvable Lie groups
spellingShingle Simply transitive NIL-affine actions of solvable Lie groups
Deré, Jonas
ALGEBRAIC HULL
LIE ALGEBRA
SEMISIMPLE SPLITTING
SIMPLY TRANSITIVE ACTION
SOLVABLE LIE GROUP
title_short Simply transitive NIL-affine actions of solvable Lie groups
title_full Simply transitive NIL-affine actions of solvable Lie groups
title_fullStr Simply transitive NIL-affine actions of solvable Lie groups
title_full_unstemmed Simply transitive NIL-affine actions of solvable Lie groups
title_sort Simply transitive NIL-affine actions of solvable Lie groups
dc.creator.none.fl_str_mv Deré, Jonas
Origlia, Marcos Miguel
author Deré, Jonas
author_facet Deré, Jonas
Origlia, Marcos Miguel
author_role author
author2 Origlia, Marcos Miguel
author2_role author
dc.subject.none.fl_str_mv ALGEBRAIC HULL
LIE ALGEBRA
SEMISIMPLE SPLITTING
SIMPLY TRANSITIVE ACTION
SOLVABLE LIE GROUP
topic ALGEBRAIC HULL
LIE ALGEBRA
SEMISIMPLE SPLITTING
SIMPLY TRANSITIVE ACTION
SOLVABLE LIE GROUP
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Every simply connected and connected solvable Lie group G admits a simply transitive action on a nilpotent Lie group H via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups G can act simply transitively on which Lie groups H. So far, the focus was mainly on the case where G is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action : G Aff(H) is simply transitive by looking only at the induced morphism p: G → aff(h) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group G acts simply transitively on a given nilpotent Lie group H, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull,whichwe also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.
Fil: Deré, Jonas. Katholikie Universiteit Leuven; Bélgica
Fil: Origlia, Marcos Miguel. Monash University; Australia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
description Every simply connected and connected solvable Lie group G admits a simply transitive action on a nilpotent Lie group H via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups G can act simply transitively on which Lie groups H. So far, the focus was mainly on the case where G is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action : G Aff(H) is simply transitive by looking only at the induced morphism p: G → aff(h) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group G acts simply transitively on a given nilpotent Lie group H, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull,whichwe also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.
publishDate 2021
dc.date.none.fl_str_mv 2021-09-01
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/172726
Deré, Jonas; Origlia, Marcos Miguel; Simply transitive NIL-affine actions of solvable Lie groups; De Gruyter; Forum Mathematicum; 33; 5; 1-9-2021; 1349-1367
0933-7741
1435-5337
CONICET Digital
CONICET
url http://hdl.handle.net/11336/172726
identifier_str_mv Deré, Jonas; Origlia, Marcos Miguel; Simply transitive NIL-affine actions of solvable Lie groups; De Gruyter; Forum Mathematicum; 33; 5; 1-9-2021; 1349-1367
0933-7741
1435-5337
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1515/forum-2020-0114
info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/document/doi/10.1515/forum-2020-0114/html
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/2004.12774
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv De Gruyter
publisher.none.fl_str_mv De Gruyter
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432