Outer derivation

Let $\mathfrak g$ be a Lie algebra .

Definitions

A derivation of $\mathfrak g$ is a linear map $D:\mathfrak g\to\mathfrak g$ satisfying the Leibniz rule

as in derivations of a Lie algebra . The space of all derivations is a Lie algebra $\mathrm{Der}(\mathfrak g)$ under the commutator bracket.

An inner derivation is one of the form $\mathrm{ad}_X$ for some $X\in\mathfrak g$, where $\mathrm{ad}_X(Y)=[X,Y]$ (see inner derivations and the adjoint representation ). The set of inner derivations is an ideal $\mathrm{Inn}(\mathfrak g)=\mathrm{ad}(\mathfrak g)\subseteq \mathrm{Der}(\mathfrak g)$.

A derivation is called an outer derivation if it is not inner. The quotient Lie algebra

measures outer derivations “modulo inner ones.”

Basic remarks

• If $X$ lies in the center of $\mathfrak g$, then $\mathrm{ad}_X=0$, so the map $\mathrm{ad}:\mathfrak g\to \mathrm{Der}(\mathfrak g)$ factors through $\mathfrak g/Z(\mathfrak g)$.
• For many rigid Lie algebras (notably semisimple ones), every derivation is inner, so the outer derivation quotient vanishes. This is one conceptual reason semisimple Lie algebras have very small deformation theory.

Context

Outer derivations appear naturally when studying extensions and deformations of Lie algebras; inner derivations encode changes coming from conjugation, while outer derivations capture genuinely new infinitesimal symmetries.