Given a simply connected solvable Lie group G, there always exists a NIL-affine action \(\rho : G \rightarrow \text {Aff}(H)\) 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 \(\rho : G \rightarrow \text {Aff}({\mathbb R}^n)\) 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 question how to define completeness in higher nilpotency classes remains open, as we illustrate with an example in the 3-step nilpotent case.
