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

Similar facts

Converse

Characteristic not implies derivation-invariant

Facts used

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 contained in $Z (L)$ , 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:

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

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 $d_{12}$ takes values 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 $L$ .