Derivation of a Lie algebra

Let $\mathfrak g$ be a Lie algebra over a field of characteristic $0$ (typically $\mathbb R$ or $\mathbb C$).

Definition

A derivation of $\mathfrak g$ is a linear map $D:\mathfrak g\to\mathfrak g$ such that for all $x,y\in\mathfrak g$,

The space of derivations is denoted $\mathrm{Der}(\mathfrak g)$.

Lie algebra structure

With bracket given by the commutator of endomorphisms,

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

Inner vs. outer

For each $x\in\mathfrak g$, the adjoint map $\mathrm{ad}_x:\mathfrak g\to\mathfrak g$, $\mathrm{ad}_x(y)=[x,y]$, is a derivation; these are the inner derivations (see inner derivation ). The assignment $x\mapsto \mathrm{ad}_x$ is the adjoint representation .

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

Motivation

Derivations are the infinitesimal analog of Lie algebra automorphisms : if $\varphi_t$ is a smooth 1-parameter family of automorphisms with $\varphi_0=\mathrm{id}$, then $\frac{d}{dt}\big|_{t=0}\varphi_t$ is a derivation.