Derivation of a Lie ring
Let be a Lie ring. A derivation of is a map satisfying the following two conditions:
- is an endomorphism with respect to the abelian group that is the additive group of .
- satisfies the Leibniz rule for the Lie bracket:
When is given the additional structure of a Lie algebra over a field or ring , then a derivation of a Lie algebra is a derivation that is also a -module map.
The derivations of a Lie ring , themselves form a Lie ring, where the Lie bracket of two derivations is their commutator. This is termed the Lie ring of derivations of and is denoted by . Further, there is a natural homomorphism of Lie rings from to . Further information: Lie ring acts as derivations by adjoint action