Growth of finitely generated simple Lie algebras

TL;DR

This paper demonstrates that any submultiplicative increasing function comparable to a polynomial can serve as the growth function for a finitely generated simple Lie algebra, solving open problems on intermediate growth.

Contribution

It constructs finitely generated simple Lie algebras with prescribed growth functions, addressing previously unresolved questions about intermediate growth in Lie algebras.

Findings

Any submultiplicative increasing function equivalent to a polynomial can be realized as a Lie algebra growth function.

Resolved two open problems on the existence of Lie algebras with intermediate growth.

Established a method to realize specific growth behaviors in simple Lie algebras.

Abstract

We realize any submultiplicative increasing function which is equivalent to a polynomial proportion of itself as the growth function of a finitely generated simple Lie algebra. As an application, we resolve two open problems posed by Petrogradsky on Lie algebras with intermediate growth.