Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild

TL;DR

This paper demonstrates that classifying certain algebraic structures, including triples of matrices and specific Lie algebras, is as complex as classifying pairs of matrices, making these problems 'wild' and intractable.

Contribution

It establishes the equivalence of classifying these algebraic structures to the well-known difficult problem of classifying matrix pairs, highlighting their inherent complexity.

Findings

Problems are 'wild' and as hard as classifying matrix pairs.

Classifying triples of symmetric or skew-symmetric matrices is hopeless.

Classifying certain Lie algebras and associative algebras is equally complex.

Abstract

We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.