Derivation-invariant not implies characteristic
This article gives the statement and possibly, proof, of a non-implication relation between two Lie subring properties. That is, it states that every Lie subring satisfying the first Lie subring property (i.e., derivation-invariant Lie subring) need not satisfy the second Lie subring property (i.e., characteristic subring of a Lie ring)
View a complete list of Lie subring property non-implications | View a complete list of Lie subring property implications
Get more facts about derivation-invariant Lie subring|Get more facts about characteristic subring of a Lie ring
- Perfect direct factor implies derivation-invariant
- Self-centralizing direct factor implies derivation-invariant
Suppose is a non-abelian Lie ring, are isomorphic copies of , and is the direct sum . Define . Then, is derivation-invariant but not characteristic.
Proof that the subring is not characteristic
Consider the coordinate exchange automorphism that interchanges and . Under this automorphism, goes to . Since is non-abelian, it is not contained in , so the image of is not equal to it.
Proof that the subring is derivation-invariant
Consider a derivation . There exist four abelian group endomorphisms that describe , namely:
In other words:
The derivation condition states that:
We thus get:
Setting gives that is a derivation. Setting gives that is a derivation. Plugging these back in, we get:
Setting in the first equation gives that for all , implying that takes values in the center of . Similarly, setting in the second equation gives that is in the center of . In particular, this implies that:
takes values in .
Thus, . Since is derivation-invariant by fact (1), , so . Thus, is a derivation-invariant subring of .