论文标题
谎言组的多项式和水平多项式函数
Polynomial and horizontally polynomial functions on Lie groups
论文作者
论文摘要
我们将多项式函数的概念推广到谎言基团，以及在卡诺组上的水平仿射图的概念。我们修复了一个子集$ \ Mathfrak g $的子集$ s $ s lievariant vector fields $ \ mathbb g $，我们假设$ s $ lie会生成$ \ mathfrak g $。我们说$ f：\ mathbb g \ to \ mathbb r $（或更一般而言，$ \ mathbb g $上的分布）是$ s $ polynomial，如果对于所有$ x \ in s $中的所有$ x \ in \ mathbb n $中都存在$ k \ in \ mathbb n $，则迭代衍生$ x^k f $是分配的效率$ x^k f $ in Distressutions。 首先，我们表明所有$ s $ polynomial功能（以及分布）均由分析函数表示，并且，如果上一个定义中的指数$ k $在s $中独立于$ x \，则它们会形成有限的维矢量空间。 其次，如果连接了$ \ Mathbb g $，并且nilpotent我们表明$ s $ - 多物种功能是莱布曼（Leibman）的多项式函数。对于非努力组而言，相同的结果可能不是正确的。 最后，我们表明，在连接的nilpotent Lie群体中，从莱布曼（Leibman）的意义上是多项式，在指数图中是多项式，并且沿着$ \ mathfrak g $的方向的某些固定程度的混合衍生物消失是等效的概念。
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous definition is independent on $X\in S$, they form a finite-dimensional vector space. Second, if $\mathbb G$ is connected and nilpotent we show that $S$-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of $\mathfrak g$ are equivalent notions.
