The Lie algebra of polynomial vector fields on the affine space is $2$-generated

TL;DR

The paper proves that the Lie algebra of polynomial vector fields on affine space, $W_n=\textnormal{Der}(\mathbb{K}^n)$, is generated by two explicit elements $U$ and $V$. The authors construct $U=\partial/\partial z_n$ and a carefully designed $V$ and show, through repeated $\operatorname{ad}$-actions and inductive arguments, that all basis elements $f\partial/\partial z_i$ (with $f$ a monomial) lie in the Lie algebra generated by $U$ and $V$, hence $W_n=\textnormal{Lie}(U,V)$. The proof builds up from $\partial/\partial z_k$ to arbitrary $z_i^s\partial/\partial z_j$ and then to all polynomial multiples, establishing the 2-generation result; the paper also discusses completeness, noting that $U$ is complete while $V$ is not complete for $n=2$, leaving open whether two complete vector fields can generate $W_n$. The work connects to broader themes of finite generation and infinite transitivity in automorphism groups of affine spaces.

Abstract

We prove that the infinite-dimensional Lie algebra of polynomial vector fields on the affine space $\KK^n$ is generated by two explicitly given elements.

Theorems & Definitions (4)

• Theorem 3.1
• proof
• Definition 4.1
• Definition 4.2