On post-Lie Algebra Structures Coming from Simply Transitive NIL-Affine Actions

Abstract

Given a simply connected solvable Lie groupG, there always exists aNIL-affine actionρ : G → Aff(H) on a nilpotent Lie group H such that G acts simply transitively.The question whether this is always possible for H = Rn abelian was known asMilnor’s question, with a negative answer due to a counterexample of Benoist. Thiscounterexample is based on a correspondence between certain affine actions ρ : G →Aff(Rn) and left-symmetric structures on the corresponding Lie algebra g of G, wheresimply transitive actions correspond exactly to the so-called complete left-symmetricstructures. In general however, the question remains open which solvable Lie groupsG can act on which nilpotent Lie groups H. A natural candidate for a correspondenceon the Lie algebra level is the notion of post-Lie algebra structures, which form thenatural generalization of left-symmetric structures. In this paper, we show that everysimply transitive NIL-affine action of G on a nilpotent Lie group H indeed induces apost-Lie algebra structure on the pair of Lie algebras (g, h). Moreover, we discuss anew notion of completeness for these structures in the case that h is 2-step nilpotent,equivalent but different from the known definition for H = Rn. We then show thatsimply transitive actions exactly correspond to complete post-Lie algebra structuresin the 2-step nilpotent case. However, the question how to define completeness inhigher nilpotency classes remains open, as we illustrate with an example in the 3-stepnilpotent case.