MathGloss

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

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