关于源于单传递 NIL-仿射作用的 post-Lie 代数结构
On post-Lie algebras structures coming from simply transitive NIL-affine actions
AI总结：
本文证明了单连通可解 Lie 群在幂零 Lie 群上的单传递 NIL-仿射作用可诱导 post-Lie 代数结构，并在 2 步幂零情形下确立了单传递作用与完备 post-Lie 代数结构的精确对应。
摘要：
给定一个单连通可解 Lie 群 $G$，总存在一个在幂零 Lie 群 $H$ 上的 NIL-仿射作用 $ρ: G \to \operatorname{Aff}(H)$，使得 $G$ 单传递地作用。对于 $H = \mathbb{R}^n$ 为交换群时这是否总是可能的问题被称为 Milnor 问题，由于 Benoist 的一个反例，该问题有了否定的答案。该反例基于某些仿射作用 $ρ: G \to \operatorname{Aff}(\mathbb R^n)$ 与 $G$ 对应的 Lie 代数 $\mathfrak g$ 上的左对称结构之间的对应关系，其中单传递作用恰好对应于所谓的完全左对称结构。然而，通常情况下，哪些可解 Lie 群 $G$ 可以作用于哪些幂零 Lie 群 $H$ 的问题仍然悬而未决。Lie 代数层面上对应关系的一个自然候选者是 post-Lie 代数结构的概念，它构成了左对称结构的自然推广。在本文中，我们证明了 $G$ 在幂零 Lie 群 $H$ 上的每一个单传递 NIL-仿射作用确实在 Lie 代数对 $(\mathfrak g,\mathfrak h)$ 上诱导出一个 post-Lie 代数结构。此外，我们讨论了在 $\mathfrak h$ 为 $2$ 步幂零的情况下这些结构的一个新的完备性概念，它等价于但不同于 $H = \mathbb R^n$ 时的已知定义。然后我们证明了在 $2$ 步幂零的情况下，单传递作用恰好对应于完备的 post-Lie 代数结构。然而，在更高幂零类中如何定义完备性的问题仍然悬而未决，正如我们用 $3$ 步幂零情况下的一个例子所说明的那样。
英文摘要：
Given a simply connected solvable Lie group $G$, there always exists NIL-affine action $ρ: G \to \operatorname{Aff}(H)$ 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 $ρ: G \to \operatorname{Aff}(\mathbb R^n)$ 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.
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