# Dimensions of semi-simple matrix algebras

## TL;DR

This paper characterizes the dimensions of semi-simple subalgebras within matrix algebras, showing that as matrix size grows, nearly all integers in a certain range are realized as such dimensions.

## Contribution

It provides a comprehensive description of which integers are dimensions of semi-simple subalgebras of matrix algebras for large n, demonstrating density approaching 1.

## Key findings

- For large n, most integers up to n^2 are dimensions of semi-simple subalgebras.
- The density of such integers tends to 1 as n increases.
- Explicit bounds on the range of realizable dimensions are established.

## Abstract

For $n \geq 225$ we show that every integer of the form $n + 2m$ such that $0 \leq 2m \leq n^{2} - \frac{9}{2} n \sqrt{n}$ is the dimension of a connected semi-simple subalgebra of $\mathrm{M}_{n}(k)$, that is, a subalgebra isomorphic to a direct sum of $t$ disjoint subalgebras $\mathrm{M}_{n_{i}}(k)$, where $\sum_{i=1}^{t} n_{i} = n$. From this, we conclude that the density of integers in $[0,\ldots, n^{2}]$ which are the dimension of a semi-simple subalgebra of $\mathrm{M}_{n}(k)$ tends to $1$ as $n \rightarrow \infty$.