Simply transitive NIL-affine actions of solvable Lie groups

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 : 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.