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
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 an additive subgroup of the Lie ring and every derivation of the Lie ring sends the subring to within itself.

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 derivation $d$ of $L$, the restriction of $d$ to $A$ is a derivation of $A$.

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

In some special circumstances, any characteristic subring of a Lie ring is derivation-invariant. This happens when every derivation can be exponentiated to an automorphism of the Lie ring.

Weaker properties

• Ideal of a Lie ring: 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

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