Invariance under any set of derivations is centralizer-closed

Statement

For a single derivation

Suppose $L$ is a Lie ring, $S$ is a subring of a Lie ring, and $d$ is a derivation of $L$ such that $d(S)\subseteq S$. Let $C=C_{L}(S)$ be the centralizer of $S$. Then, $C$ is also invariant under $d$.

For a bunch of derivations

Suppose $L$ is a Lie ring, $S$ is a subring of a Lie ring, and $D$ is a set of derivations of $L$ such that $d(S)\subseteq S$ for all $d\in D$. Let $C=C_{L}(S)$ be the centralizer of $S$. Then, $C$ is also invariant under all $d\in D$.

Related facts

Particular cases

• Ideal property is centralizer-closed
• Derivation-invariance is centralizer-closed

Other related facts

• Centralizer of a Lie subring is a Lie subring

Analogues in group theory

• Auto-invariance implies centralizer-closed
• Normality is centralizer-closed
• Characteristicity is centralizer-closed

Proof

Note that the statements for a single derivatoin and for a bunch of derivations are clearly equivalent, so we only prove the former.

Given: A Lie ring $L$, a subring $S$ of $L$. $C=C_{L}(S)$ is the set of $a\in L$ such that $[a,s]=0$ for all $s\in S$. A derivation $d$ of $L$ such that $d(S)\subseteq S$.

To prove: $d(C)\subseteq C$.

Proof: For any $c\in C$ and $s\in S$, we need to show that $[dc,s]=0$.

For this, note that, by the Leibniz rule property of derivations:

$d([c,s])=[dc,s]+[c,ds]$.

Since $[c,s]=0$, the left side is zero. Further, since $d(S)\subseteq S$, $ds\in S$, so $[c,ds]=0$. This gives $[dc,s]=0$ as required.