Integrable generators of Lie algebras of vector fields on $\mathbb{C}^n$

TL;DR

This paper constructs three specific vector fields on complex n-space whose flows generate the entire Lie algebra of algebraic vector fields with complete polynomial flows, leading to an infinitely transitive group action.

Contribution

It introduces three explicit vector fields on b^n that generate the full Lie algebra of algebraic vector fields with complete polynomial flows, a novel explicit construction.

Findings

The three vector fields have complete polynomial flows.

Their flows generate an infinitely transitive group action.

The results extend to the holomorphic setting.

Abstract

There exist three vector fields with complete polynomial flows on $\mathbb{C}^n$, $n \geq 2$, which generate the Lie algebra generated by all algebraic vector fields on $\mathbb{C}^n$ with complete polynomial flows. In particular, the flows of these vector fields generate a group that acts infinitely transitive. The analogous result holds in the holomorphic setting.