Derivation of a Lie ring

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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

Related notions