A Family of Maximal Mathieu Subspaces of Matrix Algebras

TL;DR

This paper constructs a family of maximal Mathieu subspaces within matrix algebras over a field and classifies Mathieu subspaces of 2x2 matrices when the field is algebraically closed.

Contribution

It introduces a new construction method for maximal Mathieu subspaces in matrix algebras and provides a classification for 2x2 cases over algebraically closed fields.

Findings

Constructed a family of maximal Mathieu subspaces in $M_n(F)$ for $n \\ge 2$

Classified Mathieu subspaces of $M_2(F)$ over algebraically closed fields

Abstract

Let $F$ be a field. In this note we give a construction for a family of maximal Mathieu subspaces (or Mathieu-Zhao subspaces) of the matrix algebras $M_n(F)$ $(n\ge 2)$. As an application we also give a classification of Mathieu subspaces of $M_2(F)$ under the condition that $F$ is algebraically closed.