Abelian Lie ring: Difference between revisions

Latest revision as of 23:36, 26 July 2013

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
View a complete list of properties of Lie rings
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: abelian group
View other analogues of abelian group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)

Definition

An abelian Lie ring is a Lie ring satisfying the following equivalent conditions:

The Lie bracket of any two elements is zero.
Every Lie subring of the Lie ring is an ideal in the Lie ring.

Equivalence of definitions

Further information: Lie ring is abelian iff every subring is an ideal

Relation with properties in related algebraic structures

Lie algebra

An abelian Lie algebra is an abelian Lie ring that is also a Lie algebra.

Ring whose commutator operation is the Lie bracket

Suppose $R$ is an associative ring. $R$ can be viewed as a Lie ring with the Lie bracket as $[x,y]=xy-yx$ . The Lie ring $R$ is an abelian Lie ring if and only if $R$ is a commutative ring.

Group via the Lazard correspondence

Suppose $G$ is a Lazard Lie group and $L$ is its Lazard Lie ring. $L$ is an abelian Lie ring if and only if $G$ is an abelian group.

Moreover, under the natural bijection from $L$ to $G$ , abelian subrings of $L$ correspond to abelian subgroups of $G$ .

Relation with other properties

Stronger properties

Cyclic Lie ring

Weaker properties

@@ Line 4: / Line 4: @@
 old generic context = group|
 old specific context = group|
-old property = Abelian group|
+old property = abelian group|
 new generic context = Lie ring|
 new specific context = Lie ring}}
@@ Line 10: / Line 10: @@
 ==Definition==
-An '''Abelian Lie ring''' is a [[Lie ring]] where the Lie bracket of any two elements is zero.
+An '''abelian Lie ring''' is a [[Lie ring]] satisfying the following equivalent conditions:
+# The Lie bracket of any two elements is zero.
+# Every [[Lie subring]] of the Lie ring is an [[ideal of a Lie ring|ideal]] in the Lie ring.
+===Equivalence of definitions===
+{{further|[[Lie ring is abelian iff every subring is an ideal]]}}
+==Relation with properties in related algebraic structures==
+===Lie algebra===
+An [[abelian Lie algebra]] is an abelian Lie ring that is also a [[Lie algebra]].
+===Ring whose commutator operation is the Lie bracket===
+Suppose <math>R</math> is an associative ring. <math>R</math> can be viewed as a Lie ring with the Lie bracket as <math>[x,y] = xy - yx</math>. The Lie ring <math>R</math> is an abelian Lie ring if and only if <math>R</math> is a commutative ring.
+===Group via the Lazard correspondence===
+Suppose <math>G</math> is a [[Lazard Lie group]] and <math>L</math> is its [[Lazard Lie ring]]. <math>L</math> is an abelian Lie ring if and only if <math>G</math> is an [[abelian group]].
+Moreover, under the natural bijection from <math>L</math> to <math>G</math>, abelian subrings of <math>L</math> correspond to abelian subgroups of <math>G</math>.
 ==Relation with other properties==
+===Stronger properties===
+* [[Weaker than::Cyclic Lie ring]]
 ===Weaker properties===