2-Engel Lie ring: Difference between revisions

Revision as of 07:22, 17 May 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-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.
2	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 $a \neq b$ ).
3	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.
4	2-bi-Engel Lie ring	For any $u, x, y \in L$ , we have $[u, [u, x]] = 0$ (the 2-Engel condition) and $[[u, x], [u, y]] = 0$ (the (1,1)-bi-Engel condition).

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	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Engel Lie ring				\|FULL LIST, MORE INFO

Facts

@@ Line 16: / Line 16: @@
 | 1 || 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 || 2-Engel identity || for any <math>x,y \in L</math>, we have <math>[x,[x,y]] = 0</math> (Note that if <math>x = y</math>, this would follow automatically, so we can restrict attention to the case <math>x \ne y</math>).
+| 2 || 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>).
 |-
 | 3 || 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.
+|-
+| 4 || 2-bi-Engel Lie ring || For any <math>u,x,y \in L</math>, we have <math>[u,[u,x]] = 0</math> (the 2-Engel condition) and <math>[[u,x],[u,y]] = 0</math> (the [[(1,1)-bi-Engel Lie ring|(1,1)-bi-Engel condition]]).
 |}