Inner derivation - Knowlpedia

Let $\mathfrak g$ be a Lie algebra. Recall that a derivation is a linear map $D:\mathfrak g\to\mathfrak g$ satisfying the Leibniz rule

D([x,y])=[D(x),y]+[x,D(y)].

Definition (Inner derivation). For each $x\in\mathfrak g$ , the map

\mathrm{ad}_x:\mathfrak g\to\mathfrak g,\qquad \mathrm{ad}_x(y)=[x,y]

is a derivation. A derivation is called inner if it equals $\mathrm{ad}_x$ for some $x\in\mathfrak g$ .

The assignment $x\mapsto \mathrm{ad}_x$ is the adjoint representation $\mathrm{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g)$ , and the space of inner derivations is $\mathrm{ad}(\mathfrak g)\subseteq \mathrm{Der}(\mathfrak g)$ .

Remarks

Key relation to the center. The kernel of $\mathrm{ad}$ is exactly the center (see the kernel-of-ad lemma), so inner derivations detect noncentral directions.

Context. Derivations modulo inner derivations measure “outer” symmetries of $\mathfrak g$ (compare outer derivations), and exponentiating $\mathrm{ad}_x$ is the infinitesimal source of many automorphisms.