Ideal growth in metabelian Lie P-algebras

Abstract

Consider a finitely generated restricted Lie algebra L over the finite field Fq and, given n ≥ 0, denote the number of restricted ideals H ⊂ L with $${\dim _{{F_q}}}$$ L/H = n by cn(L). We show for the free metabelian restricted Lie algebra L of finite rank that the ideal growth sequence grows superpolynomially; namely, there exist positive constants λ1 and λ2 such that $${q^{{\lambda _1}{n^2}}} \leqslant {c_n}\left( L \right) \leqslant {q^{{\lambda _2}{n^2}}}$$ for n large enough.