LAUSR

Abstract Every simply connected and connected solvable Lie group ???? admits a simply transitive action on a nilpotent Lie group ???? via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups ???? can act simply transitively on which Lie groups ????. So far, the focus was mainly on the case where ???? 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)\rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ:g→aff⁡(h)\varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group ???? acts simply transitively on a given nilpotent Lie group ????, 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.