Growth of subalgebras and subideals in free Lie algebras

TL;DR

This paper explores the growth patterns of subalgebras and subideals in free Lie algebras, revealing significant differences in their behavior using growth and cogrowth functions.

Contribution

It introduces a detailed analysis of subalgebra and subideal growth in free Lie algebras, highlighting contrasting exponential behaviors.

Findings

Proper finitely generated subalgebras have exponentially small growth.

Nonzero subideals exhibit exponentially small cogrowth.

Distinct growth behaviors differentiate subalgebras from subideals.

Abstract

We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals. For instance, the \textit{growth} of a proper finitely generated subalgebra $H$ of a free Lie algebra $L$, with respect to any fixed free basis $X$, is exponentially small compared to the growth of the whole of $L$. Quite opposite, the \textit{cogrowth} of any nonzero subideal $S$ is exponentially small compared to the growth of $L$.