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

Abstract

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 p 3 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.

Congruence of matrix spaces, matrix tuples, and multilinear maps

“…Some classes of matrix 2-tuples are classified up to weak equivalence in [2,3,9]. By [4,5], the problem of classifying matrix 3-tuples up to weak equivalence is wild, and so it contains the problems of classifying each system of linear maps and representations of each finite dimensional algebra; see [6] and [1, Proposition 9.14].…”

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

“…• finite-dimensional Lie algebras over P with central commutator subalgebra of dimension 3 (see [4,3]); • local commutative associative algebras over P with zero cube radical (see [2]); • finite p-groups of exponent p with central commutator subgroup of order p 3 (see [21]). …”

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

“…According to Belitskii and Sergeichuk [4, Section 1], wild problems are hopeless in a certain sense. Several classification problems were pointed out to be wild (see [1,2,3,4,9,20] and references therein).…”

“…Unfortunately, the problem of classifying solvable Lie algebras is wild. Indeed, Belitskii et al [1,Theorem 4] proved that the problem of classifying 2-step nilpotent Lie algebras (over an algebraically closed field of characteristic other than two) with 3-dimensional derived algebras is wild. Then so is the problem of classifying all nilpotent Lie algebras.…”