# 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 propertiesVIEW 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)

## Contents

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

- is an additive subgroup of , and for every derivation of , .
- is a Lie subring of , and for every derivation of , .
- is a Lie subring of , and for every differential operator on , .
- is a Lie subring of and every differential operator of restricts to a differential operator of .
- is a Lie subring of , and for every derivation of , the restriction of to is a derivation of .
- is an additive subgroup of , and for every differential operator on , .
- is an additive subgroup of and every differential operator of restricts to a differential operator of .

### 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 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 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`