Dimensions of semi-simple matrix algebras

Abstract

For $n\ge 225$ we show that every integer of the form $n+2m$ such that $0\le 2m\le n^2-\frac{9}{2}n\sqrt{n}$ 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 $[0,\dots ,n^2]$ which are the dimension of a semi-simple subalgebra of ${\mathrm{M}}_n(k)$ tends to $1$ as $n\to \infty$.