Invariance under any set of derivations is centralizer-closed

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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

Other related facts

Analogues in group theory

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.