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 in terms of inner derivations
Definition in terms of Lie brackets
A Lie ring is termed a -Engel Lie ring if there exists a natural number such that for any , t:
where occurs times.
Note that we sometimes simply use the term Engel Lie ring for a Lie ring where the value of may be dependent on and . However, by Zelmanov's theorem on Engel Lie rings, this is equivalent to being a locally nilpotent Lie ring.
Relation with other 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|
- Engel's theorem: This states that if 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 is a -generator Lie algebra over a field of characteristic satisfying the -Engel condition for (or arbitrary in the case ) then is nilpotent with nilpotency class bounded by a function of and .