Derivation-invariant Lie subring

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.

Metaproperties
A derivation-invariant subring of a derivation-invariant subring is again a derivation-invariant subring.

The Lie bracket of two derivation-invariant Lie subrings is again a derivation-invariant Lie subring.

The centralizer of a derivation-invariant Lie subring is again derivation-invariant.

An intersection of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant.

A join of derivation-invariant Lie subrings of a Lie ring is again derivation-invariant.