Lie algebra automorphism - Imported Knowlpedia Content

Let $\mathfrak g$ be a Lie algebra.

Definition (Automorphism). A Lie algebra automorphism is a linear isomorphism $\phi:\mathfrak g\to\mathfrak g$ such that

\phi([x,y])=[\phi(x),\phi(y)]\quad\text{for all }x,y\in\mathfrak g.

Equivalently, $\phi$ is an isomorphism in the category of Lie algebras (compare Lie algebra isomorphism).

The set $\mathrm{Aut}(\mathfrak g)$ is a group under composition; for many $\mathfrak g$ it carries a natural Lie group structure as a closed subgroup of $\mathrm{GL}(\mathfrak g)$ .

Remarks

Inner vs outer features. The adjoint representation gives canonical automorphisms by exponentiating inner derivations: for $x\in\mathfrak g$ , the map $\exp(\mathrm{ad}_x)$ is an automorphism (built from inner derivations and exponentials in $\mathrm{GL}(\mathfrak g)$ ). Automorphisms not generated this way are “outer” in nature, and are related to outer derivations at the infinitesimal level.

Context. Automorphisms organize symmetry of structure constants, control conjugacy of subalgebras (including Levi factors in the Levi decomposition), and interact with representation theory through transport of structure.