Adjoint representation of a Lie group

Summary

The adjoint representation of a Lie group $G$ is an action of $G$ on its own Lie algebra $\mathfrak{g}$. It comes from differentiating conjugation at the identity. It is a representation $\operatorname{Ad} \colon G \to GL(\mathfrak{g})$ that every Lie group carries. Its derivative is the map $X \mapsto [X, \cdot\,]$ on the Lie algebra, and its kernel, for a connected group, is the centre of $G$. It is the starting point for root systems, the Killing form and the classification of semisimple groups.