Free subalgebras of Lie algebras close to nilpotent

TL;DR

This paper demonstrates that certain Lie algebras associated with automata algebras of exponential growth contain free subalgebras, revealing structural properties near nilpotency.

Contribution

It establishes the existence of free subalgebras in Lie algebras linked to automata algebras and groups with specific commutator relations, extending understanding of their algebraic structure.

Findings

Lie algebras from automata of exponential growth contain free subalgebras

Constructs specific Lie algebras L_{n+2} with free subalgebras

Shows similar results for groups with commutator relations

Abstract

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations.