This article is cited in 1 paper

Ideal growth in metabelian Lie $p$-algebras

V. M. Petrogradsky^a, I. A. Subbotin<sup>b

a</sup> Department of Mathematics, University of Brasilia, Brasilia, Brazil
^b Faculty of Mathematics and Information Technologies, Ulyanovsk State University, Ulyanovsk, Russia

Abstract: Consider a finitely generated restricted Lie algebra $L$ over the finite field $\mathbb F_q$ and, given $n\ge0$, denote the number of restricted ideals $H\subset L$ with $\dim_{\mathbb F_q}L/H=n$ by $c_n(L)$. 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 $\lambda_1$ and $\lambda_2$ such that $q^{\lambda_1n^2}\le c_n(L)\le q^{\lambda_2n^2}$ for $n$ large enough.

Keywords: restricted Lie algebra, metabelian Lie algebra, enumerative combinatorics, subgroup growth, subalgebra growth, ideal growth.

UDC: 512.55

Received: 06.10.2014

DOI: 10.17377/smzh.2015.56.413

 English version: Siberian Mathematical Journal, 2015, 56:4, 714–724

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