Dimensions of semi-simple matrix algebras

Phillip Heikoop; Padraig Ó Catháin

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arXiv · 1905.03878

Dimensions of semi-simple matrix algebras

Phillip Heikoop·Padraig Ó Catháin

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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 $n \rightarrow \infty$.

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Phillip Heikoop, Padraig Ó Catháin. 2019-08-16. Dimensions of semi-simple matrix algebras. https://arxiv.org/abs/1905.03878

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