Simply transitive NIL-affine actions of solvable Lie groups
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/172726 |
| Acceso en línea: | http://hdl.handle.net/11336/172726 |
| Access Level: | acceso abierto |
| Palabra clave: | ALGEBRAIC HULL LIE ALGEBRA SEMISIMPLE SPLITTING SIMPLY TRANSITIVE ACTION SOLVABLE LIE GROUP https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | 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. |
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