# Derivation of a Lie ring

## Definition

Let $L$ be a Lie ring. A derivation of $L$ is a map $d:L \to L$ satisfying the following two conditions:

• $d$ is an endomorphism with respect to the abelian group that is the additive group of $L$.
• $d$ satisfies the Leibniz rule for the Lie bracket:

$d[x,y] = [dx,y] + [x,dy]$.

When $L$ is given the additional structure of a Lie algebra over a field or ring $R$, then a derivation of a Lie algebra is a derivation that is also a $R$-module map.

The derivations of a Lie ring $L$, themselves form a Lie ring, where the Lie bracket of two derivations is their commutator. This is termed the Lie ring of derivations of $L$ and is denoted by $\operatorname{Der}(L)$. Further, there is a natural homomorphism of Lie rings from $L$ to $\operatorname{Der}(L)$. Further information: Lie ring acts as derivations by adjoint action