Narrow positively graded Lie algebras

TL;DR

This paper classifies infinite-dimensional narrow positively graded Lie algebras with specific growth and dimension constraints, using central extensions of Carnot algebras, relevant in geometry and control theory.

Contribution

It provides a complete classification of such Lie algebras, highlighting their structure and connection to Carnot algebras in geometric contexts.

Findings

Classification of infinite-dimensional narrow positively graded Lie algebras.

Use of successive central extensions of finite-dimensional Carnot algebras.

Relevance to sub-Riemannian geometry, control theory, and geometric group theory.

Abstract

We classify real and complex infinite-dimensional narrow positively graded Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ In the proof of the main theorem we apply successive central extensions of finite-dimensional Carnot algebras. In sub-Riemannian geometry, control theory, and geometric group theory, Carnot algebras play a significant role.