On Zel'manov's global nilpotence theorem for Engel Lie algebras

TL;DR

This paper provides a detailed proof of Zel'manov's theorem, establishing that n-Engel Lie algebras over fields of characteristic zero are globally nilpotent, extending previous local nilpotency results.

Contribution

The paper offers a comprehensive proof of Zel'manov's global nilpotence theorem for Engel Lie algebras, clarifying his original ideas and methods.

Findings

Proves that n-Engel Lie algebras over characteristic zero fields are globally nilpotent

Extends Kostrikin's local nilpotency theorem to global nilpotency

Provides insights into Zel'manov's original proof techniques

Abstract

I give a proof of Zel'manov's theorem that if $L$ is an $n$-Engel Lie algebra over a field $F$ of characteristic zero then $L$ is (globally) nilpotent. This is a very important result which extends Kostrikin's theorem that $L$ is locally nilpotent if the characteristic of $F$ is zero or some prime $p>n$. Zel'manov's proof contains some striking original ideas, and I wrote this note in an effort to understand his arguments. I hope that my efforts will be of use to other mathematicians in understanding this remarkable theorem.