Let $G$ be a Lie group with associated Lie algebra $\mathfrak g$. Then the adjoint map $\text{Ad}:G \to \text{GL}(\mathfrak g)$ is given by \(\text{Ad}_A(X) = AXA^{-1}.\) Because the adjoint map is a Lie algebra homomorphism, it is a representation of $\mathfrak g$, and therefore we may refer to $\text{ad}$ as the adjoint representation of $G$ acting on $\mathfrak g$ when considered as a vector space.
Wikidata ID: Q1017106