论文标题
在谎言组的通用子空间上
On universal subspaces for Lie groups
论文作者
论文摘要
让$ u $为$ \ mathbb r $或$ \ mathbb c $的有限尺寸矢量空间，然后让$ρ：g \ to gl（u）$为连接的Lie Group $ G $的代表。如果$ g $的每个轨道都符合$ v $，则线性子空间$ v \ subset U $被称为通用。我们研究谎言组的通用子空间，尤其是紧凑的谎言组。 Jinpeng和Doković通过一定的拓扑阻塞与紧凑型群体的普遍性接触。他们表明，障碍物类别的不变足以满足$ v $的通用性，并询问在某些条件下是否也有必要。 在本文中，我们表明，这个问题的答案一般是负面的，但是我们讨论了答案是积极的一些重要情况。我们表明，如果$ g $是可解决的，而$ρ：g \ to gl（u）$是一个复杂的表示，那么唯一的通用复杂子空间是$ u $本身。我们还研究了Levi子组的普遍性问题。
Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $ρ:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article, we show that the answer to the question is negative in general, but we discuss some important situations where the answer is positive. We show that if $G$ is solvable and $ρ:G\to GL(U)$ is a complex representation, then the only universal complex subspace is $U$ itself. We also investigate the question of universality for a Levi subgroup.
