Derivation of a Lie algebra

Let $\mathfrak g$ be a Lie algebra over a field. A derivation of $\mathfrak g$ is a linear map $D:\mathfrak g\to\mathfrak g$ satisfying

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

for all $x,y\in\mathfrak g$ . The space of derivations is denoted $\operatorname{Der}(\mathfrak g)$ .

Lie algebra structure

With bracket given by the commutator of endomorphisms,

[D_1,D_2] := D_1\circ D_2 - D_2\circ D_1,

the space $\operatorname{Der}(\mathfrak g)$ is a Lie subalgebra of $\mathfrak{gl}(\mathfrak g)$ .

Inner vs. outer

For each $x\in\mathfrak g$ , the adjoint map $\operatorname{ad}_x:\mathfrak g\to\mathfrak g$ , defined by $\operatorname{ad}_x(y)=[x,y]$ , is a derivation. These are the inner derivations, and $x\mapsto\operatorname{ad}_x$ is the adjoint representation.

Derivations not of the form $\operatorname{ad}_x$ are outer derivations; they measure the failure of $\operatorname{ad}(\mathfrak g)$ to exhaust all infinitesimal symmetries.

Motivation

Derivations are the infinitesimal analog of Lie algebra automorphisms: if $\varphi_t$ is a smooth one-parameter family of automorphisms with $\varphi_0=\operatorname{id}$ , then $\left.\frac{d}{dt}\right|_{t=0}\varphi_t$ is a derivation.