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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/172726
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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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1844614196164034560 |
score |
13.070432 |