Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras

Abstract

For each two-dimensional vector space V of commuting n×n matrices over a field F with at least 3 elements, we denote by V˜ the vector space of all (n+1)×(n+1) matrices of the form [A¿00] with A¿V. We prove the wildness of the problem of classifying Lie algebras V˜ with the bracket operation [u,v]:=uv-vu. We also prove the wildness of the problem of classifying two-dimensional vector spaces consisting of commuting linear operators on a vector space over a field.