自同构在Lie群单参数子群空间上的远端作用刻画
Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups
AI总结：
本文刻画了Lie群自同构在单参数子群空间上的远端作用，将其与群上作用及自同构生成子群的紧性关联，并推广至子群作用，证明共轭远端作用等价于群为紧或紧与向量群直积。
摘要：
对于连通Lie群$G$及其自同构$T$，我们考虑$T$在Sub$_G$上的作用，其中Sub$_G$是$G$的闭子群构成的紧空间，配备Chabauty拓扑。我们研究$T$在Sub$^p_G$上的作用，其中Sub$^p_G$是$G$的闭单参数子群集合在Sub$_G$中的闭包。我们将$T$在Sub$^p_G$上的作用的远端性与$T$在$G$上的作用的远端性联系起来，并在$T$在极大中心环面上远端作用且$G$不是向量群时，用$T$在Aut$(G)$中生成的闭子群的紧性来刻画该远端性。我们将这些结果推广到Aut$(G)$的某个子群的作用，并将任意闭子群${\mathcal H}$在Sub$^p_G$上的远端作用等同于${\mathcal H}$中每个元素的远端作用。此外，我们证明连通Lie群$G$通过共轭作用在Sub$^p_G$上是远端的当且仅当$G$是紧致的，或者$G$同构于一个紧群与一个向量群的直积。我们的一些结果推广了Shah和Yadav的结果。
英文摘要：
For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav.
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