# Classification of certain types of maximal matrix subalgebras

## TL;DR

This paper classifies maximal nonunital intersection subalgebras of matrix algebras over a field of characteristic zero, establishing their maximum dimension and providing a complete classification for those attaining it.

## Contribution

It introduces a complete classification of maximal nonunital intersection subalgebras in matrix algebras, extending recent work and identifying their maximum dimension.

## Key findings

- Maximum dimension of nonunital intersections is (n-1)(n-2) for n ≥ 3.
- Complete classification of nonunital intersections of maximum dimension.
- Classification of maximal unital subalgebras within parabolic subalgebras.

## Abstract

Let $M_n(K)$ denote the algebra of $n \times n$ matrices over a field $K$ of characteristic zero. A nonunital subalgebra $N \subset M_n(K)$ will be called a nonunital intersection if $N$ is the intersection of two unital subalgebras of $M_n(K)$. Appealing to recent work of Agore, we show that for $n \ge 3$, the dimension (over $K$) of a nonunital intersection is at most $(n-1)(n-2)$, and we completely classify the nonunital intersections of maximum dimension $(n-1)(n-2)$. We also classify the unital subalgebras of maximum dimension properly contained in a parabolic subalgebra of maximum dimension in $M_n(K)$.