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

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. This is the authors' version of a work that was published in Linear Algebra Appl. 407 (2005) 249-262.

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

“…Let us prove that pM 1 pAq, M 2 pBqq can be used in (4) in order to prove the wildness of the problem of classifying matrix pairs up to weak similarity. We should prove that arbitrary pairs pA, Bq and pA 1 , B 1 q of mˆm matrices are similar ðñ pM 1 pAq, M 2 pBqq and pM 1 pA 1 q, M 2 pB 1 qq are weakly similar.…”

$E_8$, the most exceptional group

“…]] ⊇ · · · eventually reaches zero, there is a unique maximal one, called the radical of L and denoted rad L. The possible such ideals rad L are unclassifiable if dim(rad L) is large enough (see [11]) so we ignore it; replacing L by L/ rad L we may assume that rad L = 0. Then L is a direct sum of simple Lie algebras, which we now describe how to analyze.…”

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]). …”