5-Engel Lie algebras II

TL;DR

This paper extends the understanding of 5-Engel Lie algebras by analyzing the nilpotency of ideals generated by elements, establishing bounds on their nilpotency class in various characteristics.

Contribution

It proves that in characteristics greater than 3, the ideal generated by an element in a 5-Engel Lie algebra is nilpotent, providing explicit bounds on its nilpotency class.

Findings

Ideals generated by elements are not necessarily nilpotent in characteristic 2 and 3.

In characteristics p > 3, these ideals are nilpotent.

Explicit bounds on the nilpotency class of these ideals are obtained.

Abstract

In my article 5-Engel algebras published on the arXiv in 2023 I proved that 5-Engel Lie algebras of characteristic zero or prime characteristic $p>7$ are nilpotent of class at most 11. In this note I investigate the ideal ID$(x)$ generated by an element $x$ in a 5-Engel Lie algebra. In characteristic 2 and 3 this ideal does not have to be nilpotent. For all primes $p>3$ I show that Id$(x)$ is nilpotent in 5-Engel Lie algebras of characteristic $p$, and I obtain explicit (best possible) bounds on the nilpotency class.