Derivation-invariant Lie subring

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:

1. It is a Lie subring of the Lie ring and every derivation of the Lie ring sends the subring to within itself.
2. It is a Lie subring of the Lie ring and every derivation of the Lie ring restricts to a derivation of the subring.
3. it is a Lie subring of the Lie ring and is invariant under every differential operator of the Lie ring.
4. 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.
5. It is an additive subgroup of the Lie ring and every derivation of the Lie ring sends the subgroup to within itself.
6. It is an additive subgroup of the Lie ring and is invariant under every differential operator of the Lie ring.
7. 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:

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

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

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

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