论文标题
低维度的Carnot组的聚宝盆
A Cornucopia of Carnot groups in Low Dimensions
论文作者
论文摘要
分层的组是那些简单地连接的谎言群体，其谎言代数承认具有特征值1的特征空间正在产生的派生。当分层组配备了左右路径距离，该路径距离相对于派生引起的自动形态均匀时，该度量空间被称为Carnot组。 Carnot组出现在几个数学环境中。要了解它们的代数结构，明确研究一些示例很有用。在这项工作中，我们提供了低维分层组的列表，表达其谎言产品，并提供左右不变的矢量场的基础，以及各自的左右不变的1形，右转矢量场的基础以及其他某些属性。我们在高达7个尺寸上展示了所有分层组，还研究了一些自由型基团的尺寸，最高为14个。
Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invariant 1-forms, a basis of right-invariant vector fields, and some other properties. We exhibit all stratified groups in dimension up to 7 and also study some free-nilpotent groups in dimension up to 14.
