A Cornucopia of Carnot Groups in Low Dimensions

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2020-08-27 DOI:10.1515/agms-2022-0138

Enrico Le Donne, F. Tripaldi

引用次数: 11

Abstract

Abstract 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.

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低维卡诺群的聚宝箱

分层群是指那些单连通李群，其李代数承认一个导数，其特征值为1的特征空间是李生的。当一个分层群具有一个左不变的路径距离，且该路径距离相对于由推导导出的自同构是齐次的，这个度量空间称为卡诺群。卡诺群出现在许多数学环境中。为了理解它们的代数结构，明确地研究一些例子是有用的。在这项工作中，我们提供了一个低维分层群的列表，表达了它们的李积，并给出了左不变向量场的一个基，以及它们各自的左不变1-形式，右不变向量场的一个基，以及其他一些性质。我们展示了所有7维以下的分层群，并研究了一些14维以下的自由幂零群。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.