On small matrix subalgebras with a trivial centralizer

TL;DR

This paper determines the minimal dimensions of subalgebras with trivial centralizers in matrix algebras over a field, showing a parity-based difference and classifying certain subalgebras.

Contribution

It establishes the minimal dimensions for such subalgebras in M_n(K) and classifies all 4-dimensional cases when n is odd.

Findings

Minimal dimension is 5 for even n

Minimal dimension is 4 for odd n

Complete classification of 4-dimensional subalgebras when n is odd

Abstract

Given an integer n greater of equal to 3, we investigate the minimal dimension of a subalgebra of M_n(K) with a trivial centralizer. It is shown that this dimension is 5 when n is even and 4 when it is odd. In the latter case, we also determine all 4-dimensional subalgebras with a trivial centralizer.