On growth of metabelian Lie algebras

TL;DR

This paper constructs finitely presented metabelian Lie algebras with intermediate growth rates, expanding understanding of algebraic growth behaviors and providing explicit examples with growth types between polynomial and exponential.

Contribution

It introduces a method to produce finitely presented metabelian Lie algebras with specific intermediate growth rates, filling a gap in the classification of algebraic growth.

Findings

Explicit examples of finitely presented metabelian Lie algebras with intermediate growth.

Calculation of growth types for these algebras.

Demonstration of growth rates of the form e^{n^{d/(d+1)}} for integer d  1.

Abstract

For any integer $d\geq 1$ we construct examples of finitely presented algebras with intermediate growth of type $[e^{n^{d/(d+1)}}]$. We produce these examples by computing the growth types of some finitely presented metabelian Lie algebras.