2024/01/04 by Jonas Deré, Deré, Jonas, Marcos Origlia +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2401.02503
openalex publication_date 2024/01/04 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28
Given a simply connected solvable Lie group G, there always exists NIL-affine action ρ: 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 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 ρ: G → Aff(\mathbb Rn) 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 Rn. 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.