# Solvable Lie Algebras of Vector Fields and a Lie's Conjecture

## TL;DR

This paper provides a geometric framework for finite-dimensional solvable Lie algebras of vector fields, confirming Lie's conjecture for these cases and extending the approach to certain infinite-dimensional algebras.

## Contribution

It introduces a local, constructive geometric description of solvable Lie algebras of vector fields and proves Lie's conjecture for these structures.

## Key findings

- Confirmed Lie's conjecture for finite-dimensional solvable Lie algebras of vector fields
- Developed a geometric description applicable to infinite-dimensional cases with nilpotent derivative ideals
- Extended the framework to include transitive and analytical solvable Lie algebras

## Abstract

We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable and transitive Lie algebras of vector fields whose derivative ideal is nilpotent can be adapted to this scheme.