Ideal growth in metabelian Lie \(p\)-algebras (Q748481)

The main result (Theorem~1) of the paper under review states the following. Let \(L\) be the free metabelian restricted Lie algebra of finite rank \(d \geq 2\) over the finite field with \(q\) elements. For \(n \geq 0\), let \(c_{n}(L)\) be the number of restricted ideals of \(L\) of codimension \(n\). Then there are positive constants \(\lambda_{1}, \lambda_{2}\) such that \(q^{\lambda_{1} n^{2}} \leq c_{n}(L) \leq q^{\lambda_{2} n^{2}}\). This is an analogue of a result of \textit{D.~Segal} on the normal subgroup growth of finitely generated metabelian groups [J.\ Lond.\ Math.\ Soc., II.\ Ser.~56, No.~2, 245--263 (1997; Zbl 0923.13013)].