论文标题
可解决的谎言组的简单及时尼尔及
Simply transitive NIL-affine actions of solvable Lie groups
论文作者
论文摘要
每个简单的连接和连接的可解决的谎言组$ g $都可以通过仿射转换对nilpotent lie group $ h $进行简单的传递动作。尽管存在保证，但对哪些谎言组$ g $可以简单地采取哪些谎言组$ h $的行为并不了解。到目前为止，重点主要放在$ g $也是nilpotent的情况下，仅取决于相应的谎言代数，并且与lie后代数结构的概念有关。 本文研究了这个问题的两个不同方面。首先，我们提供了一种方法来检查给定的动作$ρ：g \ to \ operatatorName {aff}（h）$仅通过仅查看诱导的某些词法$φ：\ mathfrak {g} \ to \ operatatorNorname {aff}（\ m athfrak {h}）$之间的均传递。其次，我们展示了如何通过研究相应的Lie代数的属性来检查给定的可解决的谎言组$ g $ ACT是否只是在给定的nilpotent Lie Group $ H $上进行转移。两种方法的主要工具是可解决的谎言代数的半神经分裂及其与代数船体的关系，我们也在lie代数的级别上定义了该代数。作为一个应用程序，我们全面说明了简单的及时操作的可能性，直至尺寸$ 4 $。
Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $ρ: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $φ: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$.
