Engel Lie ring

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
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ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: Engel group
View other analogues of Engel group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

Definition

Definition in terms of inner derivations

A Lie ring is termed a $n$-Engel Lie ring if every inner derivation of the Lie ring is a nilpotent derivation with nilpotency at most $n$.

Definition in terms of Lie brackets

A Lie ring $L$ is termed a $n$-Engel Lie ring if there exists a natural number $n$ such that for any $x,y\in L$, t:

$[x,[x,\dots ,[x,y]\dots ]]$

where $x$ occurs $n$ times.

Note that we sometimes simply use the term Engel Lie ring for a Lie ring where the value of $n$ may be dependent on $x$ and $y$. However, by Zelmanov's theorem on Engel Lie rings, this is equivalent to being a locally nilpotent Lie ring.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent Lie ring
Lazard Lie ring
inner-Lazard Lie ring
Lie ring in which every inner derivation is exponentiable	every inner derivation is an exponentiable derivation

Facts

• Engel's theorem: This states that if $L$ is a finite-dimensional Lie algebra over a field, the Engel property is equivalent to being nilpotent.
• Zelmanov's theorem on Engel Lie rings: This states that Engel Lie rings are locally nilpotent.
• Kostrikin's theorem: This states that if $L$ is a $d$-generator Lie algebra over a field of characteristic $p$ satisfying the $n$-Engel condition for $n<p$ (or $n$ arbitrary in the case $p=0$) then $L$ is nilpotent with nilpotency class bounded by a function of $d$ and $n$.