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 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 ∈ {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≤100 as well as the asymptotic behavior of the average, ?_X(r), for large r. In particular, we prove that there exist positive constants b and c such that exp(br~(2/3))≤exp(cr~(2/3)). The analogous average for matrix algebras M_n (F) is proved to be a ln n+O(1) where a is an explicit constant depending on char F.
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机译：已知的分类结果使我们能够找到矩阵代数和经典简单李代数在代数封闭域F上的精细等级（等价类）的数量（假设在Lie情况下为char F≠2）。对于矩阵代数，特别是对于类型为B_r的简单李代数（答案仅为r + 1），该计算是容易的，但是对于系列A，C和D而言，涉及对某些有限群的轨道进行计数。对于X∈{A ，CD}，我们确定在r≤100的简单X_r型李代数上，细级数N_X（r）的确切数目，以及大r的平均值？_X（r）的渐近行为。特别地，我们证明存在正常数b和c使得exp（br〜（2/3））≤exp（cr〜（2/3））。矩阵代数M_n（F）的相似平均值被证明是ln n + O（1），其中a是取决于char F的显式常数。
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