On post-Lie algebras structures coming from simply transitive NIL-affine actions

Given a simply connected solvable Lie group $G$, there always exists NIL-affine action $\rho: G \to \operatorname{Aff}(H)$ on a nilpotent Lie group $H$ such that $G$ acts simply transitively. The question whether this is always possible for $H = \mathbb{R}^n$ abelian was known as Milnor's question, with a negative answer due to a counterexample of Benoist. This counterexample is based on a correspondence between certain affine actions $\rho: G \to \operatorname{Aff}(\mathbb R^n)$ and left-symmetric structures on the corresponding Lie algebra $\mathfrak g$ of $G$, where simply transitive actions correspond exactly to the so-called complete left-symmetric structures. In general however, the question remains open which solvable Lie groups $G$ can act on which nilpotent Lie groups $H$. A natural candidate for a correspondence on the Lie algebra level is the notion of post-Lie algebra structures, which form the natural generalization of left-symmetric structures. In this paper, we show that every simply transitive NIL-affine action of $G$ on a nilpotent Lie group $H$ indeed induces a post-Lie algebra structure on the pair of Lie algebras $(\mathfrak g,\mathfrak h)$. Moreover, we discuss a new notion of completeness for these structures in the case that $\mathfrak h$ is $2$-step nilpotent, equivalent but different from the known definition for $H = \mathbb R^n$. We then show that simply transitive actions exactly correspond to complete post-Lie algebra structures in the $2$-step nilpotent case. However, the questions how to define completeness in higher nilpotency classes remains open, as we illustrate with an example in the $3$-step nilpotent case.