Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting fine gradings on matrix algebras and on classical simple Lie
  algebras   Known classification results allow us to find the number of (equivalence
classes of) fine gradings on matrix algebras and on classical simple Lie
algebras over an algebraically closed field $\mathbb{F}$ (assuming
$\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for
matrix algebras and especially for simple Lie algebras of type $B_r$ (the
answer is just $r+1$), but involves counting orbits of certain finite groups in
the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the
exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type
$X_r$ with $r\le 100$ as well as the asymptotic behaviour of the average, $\hat
N_X(r)$, for large $r$. In particular, we prove that there exist positive
constants $b$ and $c$ such that $\exp(br^{2/3})\le\hat
N_X(r)\le\exp(cr^{2/3})$. The analogous average for matrix algebras
$M_n(\mathbb{F})$ is proved to be $a\ln n+O(1)$ where $a$ is an explicit
constant depending on $\mathrm{char} \mathbb{F}$.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th