Adjoint Action of a Lie Group

Let $G$ be a Lie group . The conjugation map by $g\in G$ is

This defines a smooth action of $G$ on itself by group automorphisms.

Induced action on the Lie algebra

Differentiating at the identity gives a linear map

called the adjoint action of $G$ on its Lie algebra. The assignment $g\mapsto \operatorname{Ad}_g$ is a Lie group homomorphism

where $\operatorname{Aut}(\mathfrak{g})$ denotes Lie algebra automorphisms .

Kernel and center

The kernel of $\operatorname{Ad}$ is the center of the group $Z(G)$: elements commuting with all of $G$.

Matrix group picture and exponentials

For a matrix Lie group $G\subseteq \operatorname{GL}(n)$,

Moreover, with the exponential map one has the fundamental relation

linking $\operatorname{Ad}$ to the adjoint representation of the Lie algebra and ultimately to the Baker–Campbell–Hausdorff formula .

Adjoint actions play a central role in structure theory (e.g. Cartan subalgebras , root systems , and the Weyl group ).