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 𝔽 (assuming char 𝔽 ≠ 2 in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type Br(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 ∈ {A, C, D}, we determine the exact number of fine gradings, NX(r), on the simple Lie algebras of type Xrwith r ≤ 100 as well as the asymptotic behavior of the average, [Formula: see text], for large r. In particular, we prove that there exist positive constants b and c such that [Formula: see text]. The analogous average for matrix algebras Mn(𝔽) is proved to be a ln n + O(1) where a is an explicit constant depending on char 𝔽.
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