Enumeration of maximal subalgebras in free restricted Lie algebras

Given a finitely generated restricted Lie algebra $L$ over the finite field $\mathbb F_q$, and $n\ge0$, denote by $a_n(L)$ the number of restricted subalgebras $H\subseteq L$ with $\dim_{\mathbb F_q}L/H=n$. Denote by $\widetilde a_n(L)$ the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra $L=F_d$ of rank $d\ge2$, we find the asymptotics of $\widetilde a_n(F_d)$ and show that it coincides with the asymptotics of $a_n(F_d)$ which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras $H\subset F_d$ of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.