Engel Lie ring

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This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
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VIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions
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.