Derivation-invariant Lie subring: Difference between revisions

Latest revision as of 20:13, 27 June 2012

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
View a complete list of such properties
VIEW RELATED: Lie subring property implications | Lie subring property non-implications | Lie subring metaproperty satisfactions | Lie subring metaproperty dissatisfactions | Lie subring property satisfactions |Lie subring property dissatisfactions

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie subring property analogous to the subgroup property: characteristic subgroup
An alternative analogue of characteristic subgroup in Lie ring is: characteristic subring of a Lie ring
View other analogues of characteristic subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

Symbol-free definition

A subset of a Lie ring is termed a derivation-invariant Lie subring if it satisfies the following equivalent conditions:

It is a Lie subring of the Lie ring and every derivation of the Lie ring sends the subring to within itself.
It is a Lie subring of the Lie ring and every derivation of the Lie ring restricts to a derivation of the subring.
it is a Lie subring of the Lie ring and is invariant under every differential operator of the Lie ring.
It is a Lie subring of the Lie ring and every differential operator of the Lie ring restricts to a differential operator of the subring.
It is an additive subgroup of the Lie ring and every derivation of the Lie ring sends the subgroup to within itself.
It is an additive subgroup of the Lie ring and is invariant under every differential operator of the Lie ring.
It is an additive subgroup of the Lie ring and every differential operator of the Lie ring restricts to a differential operator of the subring.

Definition with symbols

A subset $A$ of a Lie ring $L$ is termed a derivation-invariant Lie subring if it satisfies the following equivalent conditions:

$A$ is an additive subgroup of $L$ , and for every derivation $d$ of $L$ , $d (A) \subseteq A$ .
$A$ is a Lie subring of $L$ , and for every derivation $d$ of $L$ , $d (A) \subseteq A$ .
$A$ is a Lie subring of $L$ , and for every differential operator $d$ on $L$ , $d (A) \subseteq A$ .
$A$ is a Lie subring of $L$ and every differential operator $d$ of $L$ restricts to a differential operator of $A$ .
$A$ is a Lie subring of $L$ , and for every derivation $d$ of $L$ , the restriction of $d$ to $A$ is a derivation of $A$ .
$A$ is an additive subgroup of $L$ , and for every differential operator $d$ on $L$ , $d (A) \subseteq A$ .
$A$ is an additive subgroup of $L$ and every differential operator $d$ of $L$ restricts to a differential operator of $A$ .

More general notion for non-associative rings

For the more general notion, see derivation-invariant subring of a non-associative ring.

Formalisms

Template:Lie-frexp

The property of being derivation-invariant can be expressed in terms of the function restriction formalism for Lie rings in the following ways:

As the invariance property with respect to the property of being a derivation, i.e.:

Derivation $\to$ Function

In other words, any derivation of the whole Lie ring restricts to a function from the Lie subring to itself.

As the balanced property with respect to the property of being a derivation, i.e.:

Derivation $\to$ Derivation

In other words, any derivation of the whole Lie ring restricts to a derivation from the Lie subring to itself.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lie subring invariant under any derivation with partial divided Leibniz condition powers
self-derivation-invariant Lie subring

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
ideal of a Lie ring	An ideal is a subring invariant under all inner derivations.	derivation-invariant implies ideal (the proof is similar to characteristic implies normal)	ideal not implies derivation-invariant

Metaproperties

Transitivity

This Lie subring property is transitive: a Lie subring with this property in a Lie subring with this property, also has this property.
View a complete list of transitive Lie subring properties

A derivation-invariant subring of a derivation-invariant subring is again a derivation-invariant subring. For full proof, refer: Derivation-invariance is transitive

Lie brackets

This Lie subring property is Lie bracket-closed: the Lie bracket of any two Lie subrings, both with this property, also has this property.
View a complete list of Lie bracket-closed Lie subring properties

The Lie bracket of two derivation-invariant Lie subrings is again a derivation-invariant Lie subring. For full proof, refer: Derivation-invariance is Lie bracket-closed

Centralizer-closedness

This Lie subring property is centralizer-closed: the centralizer of a Lie subring with this property in the whole Lie ring also has the property in the whole Lie ring.
View a complete list of centralizer-closed Lie subring properties

The centralizer of a derivation-invariant Lie subring is again derivation-invariant. For full proof, refer: Derivation-invariance is centralizer-closed

Template:Intersection-closed Lie subring property

An intersection of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant. For full proof, refer: Derivation-invariance is strongly intersection-closed

Template:Join-closed Lie subring property

A join of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant. For full proof, refer: Derivation-invariance is strongly join-closed

@@ Line 1: / Line 1: @@
-{{LIe subring property}}
+{{Lie subring property}}
-{{analogue in-of|Lie ring|subgroup|characteristic}}
+{{analogue of property|
+new generic context = Lie ring|
+new specific context = Lie subring|
+old generic context = group|
+old specific context = subgroup|
+old property = characteristic subgroup|
+alternative analogue = characteristic subring of a Lie ring}}
 ==Definition==
-A subring <math>A</math> of a [[Lie ring]] <math>L</math> is termed '''derivation-invariant''' if <math>d(A) \subset A</math> for every derivation <math>d</math> of <math>L</math>.
+===Symbol-free definition===
+A subset of a [[Lie ring]] is termed a '''derivation-invariant Lie subring''' if it satisfies the following equivalent conditions:
+# It is a [[Lie subring]] of the Lie ring and every [[defining ingredient::derivation of a Lie ring|derivation]] of the Lie ring sends the subring to within itself.
+# It is a [[Lie subring]] of the Lie ring and every derivation of the Lie ring restricts to a derivation of the subring.
+# it is a [[Lie subring]] of the Lie ring and is invariant under every [[defining ingredient::ring of differential operators on a Lie ring|differential operator]] of the Lie ring.
+# It is a [[Lie subring]] of the Lie ring and every differential operator of the Lie ring restricts to a differential operator of the subring.
+# It is an additive subgroup of the Lie ring and every [[defining ingredient::derivation of a Lie ring|derivation]] of the Lie ring sends the subgroup to within itself.
+# It is an additive subgroup of the Lie ring and is invariant under every [[defining ingredient::ring of differential operators on a Lie ring|differential operator]] of the Lie ring.
+# It is an additive subgroup of the Lie ring and every differential operator of the Lie ring restricts to a differential operator of the subring.
+===Definition with symbols===
+A subset <math>A</math> of a [[Lie ring]] <math>L</math> is termed a '''derivation-invariant Lie subring''' if it satisfies the following equivalent conditions:
+# <math>A</math> is an additive subgroup of <math>L</math>, and for every derivation <math>d</math> of <math>L</math>, <math>d(A) \subseteq A</math>.
+# <math>A</math> is a [[Lie subring]] of <math>L</math>, and for every derivation <math>d</math> of <math>L</math>, <math>d(A) \subseteq A</math>.
+# <math>A</math> is a [[Lie subring]] of <math>L</math>, and for every differential operator <math>d</math> on <math>L</math>, <math>d(A) \subseteq A</math>.
+# <math>A</math> is a [[Lie subring]] of <math>L</math> and every differential operator <math>d</math> of <math>L</math> restricts to a differential operator of <math>A</math>.
+# <math>A</math> is a [[Lie subring]] of <math>L</math>, and for every derivation <math>d</math> of <math>L</math>, the restriction of <math>d</math> to <math>A</math> is a derivation of <math>A</math>.
+# <math>A</math> is an additive subgroup of <math>L</math>, and for every differential operator <math>d</math> on <math>L</math>, <math>d(A) \subseteq A</math>.
+# <math>A</math> is an additive subgroup of <math>L</math> and every differential operator <math>d</math> of <math>L</math> restricts to a differential operator of <math>A</math>.
+===More general notion for non-associative rings===
+For the more general notion, see [[derivation-invariant subring of a non-associative ring]].
+==Formalisms==
+{{Lie-frexp}}
+The property of being derivation-invariant can be expressed in terms of the function restriction formalism for Lie rings in the following ways:
+* As the invariance property with respect to the property of being a derivation, i.e.:
+Derivation <math>\to</math> Function
+In other words, any derivation of the whole Lie ring restricts to a function from the Lie subring to itself.
+* As the balanced property with respect to the property of being a derivation, i.e.:
+Derivation <math>\to</math> Derivation
+In other words, any derivation of the whole Lie ring restricts to a derivation from the Lie subring to itself.
 ==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Lie subring invariant under any derivation with partial divided Leibniz condition powers]] || || || ||
+|-
+| [[Weaker than::self-derivation-invariant Lie subring]] || || || ||
+|}
 ===Weaker properties===
-* [[Ideal]]: This is a subring invariant under all ''inner'' derivations. The fact that any derivation-invariant subring is an ideal, is analogous to the fact that [[characteristic implies normal]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[stronger than::ideal of a Lie ring]] || An ideal is a subring invariant under all ''inner'' derivations. || [[derivation-invariant implies ideal]] (the proof is similar to [[characteristic implies normal]]) || [[ideal not implies derivation-invariant]] ||
+|}
+==Metaproperties==
+{{transitive Lie subring property}}
+A derivation-invariant subring of a derivation-invariant subring is again a derivation-invariant subring. {{proofat|[[Derivation-invariance is transitive]]}}
+{{Lie bracket-closed Lie subring property}}
+The Lie bracket of two derivation-invariant Lie subrings is again a derivation-invariant Lie subring. {{proofat|[[Derivation-invariance is Lie bracket-closed]]}}
+{{centralizer-closed Lie subring property}}
+The centralizer of a derivation-invariant Lie subring is again derivation-invariant. {{proofat|[[Derivation-invariance is centralizer-closed]]}}
+{{intersection-closed Lie subring property}}
+An intersection of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant. {{proofat|[[Derivation-invariance is strongly intersection-closed]]}}
+{{join-closed Lie subring property}}
+A join of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant. {{proofat|[[Derivation-invariance is strongly join-closed]]}}