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

TL;DR

This paper demonstrates the complexity and unpredictability ('wildness') in classifying certain two-dimensional spaces of commuting matrices and related Lie algebras, highlighting their intrinsic mathematical difficulty.

Contribution

It proves the classification problems for these spaces and Lie algebras are 'wild', indicating they are as complex as the most difficult problems in their domain.

Findings

Classification of these spaces is wild and intractable.

The problem remains complex for fields with at least 3 elements.

The results highlight the inherent difficulty in classifying such algebraic structures.

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 $\widetilde V$ the vector space of all $(n+1)\times(n+1)$ matrices of the form $\left[\begin{smallmatrix}A&*\\0&0\end{smallmatrix}\right]$ with $A\in V$. We prove the wildness of the problem of classifying Lie algebras $\widetilde V$ 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.