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

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 $\mathbb R^{n}.$