Counting fine gradings on matrix algebras and on classical simple Lie algebras

TL;DR

This paper counts and analyzes the number of fine gradings on matrix and classical simple Lie algebras over algebraically closed fields, providing exact counts for small ranks and asymptotic growth rates for large ranks.

Contribution

It offers explicit counts for fine gradings on classical Lie algebras up to rank 100 and establishes asymptotic bounds for their average number as rank increases.

Findings

Exact counts for ranks up to 100.

Asymptotic exponential growth bounds for classical Lie algebras.

Average number of fine gradings on matrix algebras grows logarithmically with dimension.

Abstract

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 $\mathrm{char} \mathbb{F}$.