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 $n\times n$ matrices over a field $\mathbb{F}$ with at least 3 elements, we denote by $\tilde{V}$ the vector space of all $(n+1)\times (n+1)$ matrices of the form $\left[\begin{array}{cc}A& \ast \\ 0& 0\end{array}\right]$ with $A\in V$. We prove the wildness of the problem of classifying Lie algebras $\tilde{V}$ 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.

11 pages

References in corpus (3)

• Complexity of matrix problems
• Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild
• On Lie algebras associated with modules over polynomial rings

Cited by in corpus (2)

• Congruence of matrix spaces, matrix tuples, and multilinear maps
• Classification of a Class of Solvable Real Lie Algebras by using Techniques in Matrix Theory