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- Title
Simply transitive NIL-affine actions of solvable Lie groups.
- Authors
Deré, Jonas; Origlia, Marcos
- Abstract
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 ρ : G → Aff(H) is simply transitive by looking only at the induced morphism φ : 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, which we 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.
- Subjects
LIE groups; SOLVABLE groups; NILPOTENT Lie groups; LIE algebras; RELATION algebras; AFFINE transformations
- Publication
Forum Mathematicum, 2021, Vol 33, Issue 5, p1349
- ISSN
0933-7741
- Publication type
Article
- DOI
10.1515/forum-2020-0114