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)
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Definition

There can exist a Lie ring $L$ with a subring $S$ such that $S$ is a derivation-invariant Lie subring of $L$, such that $S$ is not a characteristic subring of $L$.

Related facts

• Characteristic not implies derivation-invariant

Facts used

1. Center is derivation-invariant

Proof

Suppose $A$ is a non-abelian Lie ring, $A_{1},A_{2}$ are isomorphic copies of $A$, and $L$ is the direct sum $A_{1}\oplus A_{2}$. Define $S=A_{1}+Z(L)$. Then, $S$ is derivation-invariant but not characteristic.

Proof that the subring is not characteristic

Consider the coordinate exchange automorphism that interchanges $A_{1}$ and $A_{2}$. Under this automorphism, $A_{1}+Z(L)$ goes to $A_{2}+Z(L)$. Since $A_{2}$ is non-abelian, it is not contianedin $Z(L)$, so the image of $A_{1}+Z(L)$ is not equal to it.

Proof that the subring is derivation-invariant

Consider a derivation $d:L\to L$. There exist four abelian group endomorphisms $d_{11},d_{12},d_{21},d_{22}$ that describe $d$, namely:

$d(x,0)=(d_{11}(x),d_{12}(x)),\qquad d(0,y)=(d_{21}(y),d_{22}(y))$.

In other words:

$\!d(x,y)=(d_{11}(x)+d_{21}(y),d_{12}(x)+d_{22}(y))$.

The derivation condition states that:

$\!d[(x,y),(x',y')]=[d(x,y),(x',y')]+[(x,y),d(x',y')]$.

This gives:

$\!(d_{11}([x,x'])+d_{21}([y,y']),d_{12}([x,x'])+d_{22}([y,y']))=([d_{11}(x),x']+[d_{21}(y),x']+[x,d_{11}(x')]+[x,d_{12}(y')],[d_{12}(x),y']+[d_{22}(y),y'])+[y,d_{21}(x')]+[y,d_{22}(y')])$.

We thus get:

$\!d_{11}([x,x'])+d_{21}([y,y'])=[d_{11}(x),x']+[d_{21}(y),x']+[x,d_{11}(x')]+[x,d_{12}(y')]$

and:

$\!d_{12}([x,x'])+d_{22}([y,y'])=[d_{12}(x),y']+[d_{22}(y),y'])+[y,d_{21}(x')]+[y,d_{22}(y')]$.

Setting $y=y'=0$ gives that $d_{11}$ is a derivation. Setting $x=x'=0$ gives that $d_{22}$ is a derivation. Plugging these back in, we get:

$\!d_{21}([y,y'])=[d_{21}(y),x']+[x,d_{12}(y')]$

and:

$\!d_{12}([x,x'])=[d_{12}(x),y']+[y,d_{21}(x')]$.

Setting $y'=0$ in the first equation gives that $[d_{21}(y),x']=0$ for all $x',y$, implying that $d_{21}$ takes values in the center of $A_{1}$. Similarly, setting $x'=0$ in the second equation gives that $d_{12}(x)$ is in the center of $A_{2}$. In particular, this implies that:

$\!d(x,0)=(d_{11}(x),d_{12}(x))$

takes values in $A_{1}\oplus Z(A_{2})=A_{1}+Z(L)=S$.

Thus, $d(A_{1})\subseteq S$. Since $Z(L)$ is derivation-invariant by fact (1), $d(Z(L))\subseteq Z(L)$, so $d(S)=d(A_{1})+d(Z(L))\subseteq S+Z(L)=S$. Thus, $S$ is a derivation-invariant subring of $L$.