COUNTING FINE GRADINGS ON MATRIX ALGEBRAS AND ON CLASSICAL SIMPLE LIE ALGEBRAS

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 Xr with 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.