Wildness of the problem of classifying nilpotent Lie algebras of vector fields in four variables

TL;DR

This paper demonstrates that classifying finite-dimensional subalgebras of the Lie algebra of polynomial vector fields in four variables is a complex, 'wild' problem, indicating it contains the difficulty of classifying matrix pairs up to similarity.

Contribution

The paper proves that the classification problem for finite-dimensional subalgebras of polynomial vector fields in four variables is wild, extending known results from lower dimensions.

Findings

Classification is solved for n ≤ 2 over ℂ or ℝ.

For n ≥ 4, the problem is wild, containing the matrix pair classification.

Finite-dimensional subalgebras relate to polynomial vector fields on ℝⁿ.

Abstract

Let $\mathbb F$ be a field $\mathbb F $ of characteristic zero. Let $W_{n}(\mathbb F)$ be the Lie algebra of all $\mathbb F$-derivations with the Lie bracket $[D_1, D_2]:=D_1D_2-D_2D_1$ on the polynomial ring $\mathbb F [x_1, \ldots , x_n]$. The problem of classifying finite dimensional subalgebras of $W_{n}(\mathbb F)$ was solved if $ n\leq 2$ and $\mathbb F=\mathbb C$ or $\mathbb F=\mathbb R.$ We prove that this problem is wild if $n\geq 4$, which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity. The structure of finite dimensional subalgebras of $W_{n}(\mathbb F)$ is interesting since each derivation in case $\mathbb F=\mathbb R$ can be considered as a vector field with polynomial coefficients on the manifold $\mathbb R^{n}.$