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

Abstract

Let F be a field of characteristic zero. Let Wn(F) be the Lie algebra of all F-derivations with the Lie bracket [D1,D2]:=D1D2−D2D1 on the polynomial ring F[x1,…,xn]. The problem of classifying finite dimensional subalgebras of Wn(F) was solved if n⩽2 and F=C or F=R. We prove that this problem is wild if n⩾4, which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity. The structure of finite dimensional subalgebras of Wn(F) is interesting since each derivation in case F=R can be considered as a vector field with polynomial coefficients on the manifold Rn.