Derivation of a Lie algebra - Imported Knowlpedia Content

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

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

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

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

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 $\mathrm{Der}(\mathfrak g)$ 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.