Inner derivation of a Lie ring

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a derivation property analogous to the automorphism property: inner automorphism
View other analogues of inner automorphism | View other analogues in Lie rings of automorphism properties (OR, View as a tabulated list)

Definition

Let $L$ be a Lie ring. Given $x\in L$, the inner derivation of $L$ induced by $x$, denoted by $adx$, is the map $y\mapsto [x,y]$.

The inner derivation induced by $x$ is also called the adjoint action of $x$ or the adjoint map induced by $x$.

Facts

• The inner derivation induced by any element is a derivation
• The set of inner derivations forms an ideal inside the ring of all derivations
• The map from a Lie ring to its Lie ring of derivations, which sends $x$ to $adx$, is a homomorphism of Lie rings. Its kernel is the center of the Lie ring, and its image is the set of inner derivations

These facts are very similar to those for the map from a group to its automorphism group, sending each element to the corresponding conjugation operation. In fact, for those Lie algebras which arise from Lie groups, there is a mapping from derivations to automorphisms, by the exponential, under which inner derivations go to inner automorphisms of the corresponding Lie group (and also yield automorphisms of the Lie algebra).