2-Engel Lie ring: Difference between revisions

Latest revision as of 00:19, 6 June 2012

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: 2-Engel group
View other analogues of 2-Engel group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

Definition

A 2-Engel Lie ring can be defined in the following equivalent ways:

No.	Shorthand	A Lie ring $L$ is termed a 2-Engel Lie ring if ...
1	2-Engel identity	for any $a, b \in L$ , we have $[a, [a, b]] = 0$ (Note that if $a = b$ , this would follow automatically, so we can restrict attention to the case $a \neq b$ ).
2	triple Lie bracket is alternating	The function $(x, y, z) \mapsto [x, [y, z]]$ is an alternating function in all pairs of variables.
3	2-locally class at most two	any subring of $L$ generated by a subset of size at most two is a Lie ring of nilpotency class two, i.e., any such subring has class at most two.
4	cyclic symmetry of Lie bracket	for any $x, y, z \in L$ , we have $[x, [y, z]] = [y, [z, x]] = [z, [x, y]]$ . Note that $x, y, z$ are possibly equal, possibly distinct.
5	union of abelian ideals	There is a collection $A_{i}, i \in I$ (for some indexing set $I$ ) of abelian ideals of $L$ such that $L = ⋃_{i \in I} A_{i}$ . Equivalently, every element of $L$ is contained in an abelian ideal.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian Lie ring	Lie bracket of any two elements is trivial			\|FULL LIST, MORE INFO
Lie ring of nilpotency class two	$[x, [y, z]] = 0$ for any (not necessarily distinct) elements $x, y, z$ of the Lie ring			\|FULL LIST, MORE INFO

Weaker properties

Property	Proof of implication	Intermediate notions
Engel Lie ring		\|FULL LIST, MORE INFO
3-Engel Lie ring
Lie ring of nilpotency class three	2-Engel implies class three for Lie rings
metabelian Lie ring	(via class three)
(1,1)-bi-Engel Lie ring	(via metabelian)
2-Engel alternating ring

Facts

@@ Line 16: / Line 16: @@
 | 1 || 2-Engel identity || for any <math>a,b \in L</math>, we have <math>[a,[a,b]] = 0</math> (Note that if <math>a = b</math>, this would follow automatically, so we can restrict attention to the case <math>a \ne b</math>).
 |-
-| 2 || 2-locally class at most two || any subring of <math>L</math> generated by a subset of size at most two is a [[defining ingredient::Lie ring of nilpotency class two]], i.e., any such subring has class at most two.
+| 2 || triple Lie bracket is alternating || The function <math>(x,y,z) \mapsto [x,[y,z]]</math> is an alternating function in all pairs of variables.
 |-
-| 3 || triple Lie bracket is alternating || The function <math>(x,y,z) \mapsto [x,[y,z]]</math> is an alternating function in all pairs of variables.
+| 3 || 2-locally class at most two || any subring of <math>L</math> generated by a subset of size at most two is a [[defining ingredient::Lie ring of nilpotency class two]], i.e., any such subring has class at most two.
 |-
 | 4 || cyclic symmetry of Lie bracket || for any <math>x,y,z \in L</math>, we have <math>[x,[y,z]] = [y,[z,x]] = [z,[x,y]]</math>. Note that <math>x,y,z</math> are possibly equal, possibly distinct.