Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild

Abstract

Let F be a field of characteristic different from 2. It is shown that the problems of
classifying
    (i) local commutative associative algebras over F with zero cube radical,
    (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and
    (iii) finite p-groups of exponent p with central commutator subgroup of order p³
are hopeless since each of them contains
     • the problem of classifying symmetric bilinear mappings U × U → V , or
     • the problem of classifying skew-symmetric bilinear mappings U × U → V ,
in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.