The maximal dimension of unital subalgebras of the matrix algebra

TL;DR

This paper determines the maximum dimension of proper unital subalgebras in matrix algebras over characteristic zero fields, showing it is achieved by parabolic subalgebras, and also explores minimal coideal dimensions in matrix coalgebras.

Contribution

It establishes the exact maximal dimension of unital subalgebras in matrix algebras and characterizes the subalgebras attaining this bound, extending to matrix coalgebras.

Findings

Maximal dimension of unital subalgebras is n^2 - n + 1.

Parabolic subalgebras attain this maximal dimension.

Minimal dimension of non-zero coideals is n-1.

Abstract

Using Wederburn's main theorem and a result of Gerstenhaber we prove that, over a field of characteristic zero, the maximal dimension of a proper unital subalgebra in the $n \times n$ matrix algebra is $n^2 - n + 1$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ${\mathcal M}^n (k)$ is $n-1$.