A Cornucopia of Carnot groups in Low Dimensions

TL;DR

This paper provides a comprehensive classification and explicit descriptions of low-dimensional Carnot groups, including their algebraic structures and invariant vector fields, to facilitate understanding of their properties.

Contribution

It explicitly lists and characterizes all stratified groups up to dimension 7 and some free-nilpotent groups up to dimension 14, enriching the algebraic understanding of these groups.

Findings

Complete list of low-dimensional stratified groups

Explicit Lie algebra structures and invariant vector fields

Analysis of free-nilpotent groups up to dimension 14

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.