[[Representation of a compact Lie group]]
# Adjoint representation

Given a [[Lie group]] $G$ with its corresponding [[Lie algebra]] $\mathfrak{g}$, we may define an **adjoint representation** $\Ad_{-} : G \to \mathrm{GL}(\mathfrak{g})$ by conjugation #m/def/lie 
$$
\begin{align*}
\Ad_{g} : \mathfrak{g} &\to\mathfrak{g} \\
X &\mapsto gXg^{-1}
\end{align*}
$$

## Induced Lie algebra representation

See [[Adjoint Lie algebra representation]].

## Properties

- $\Ad_{e^X} = e^{\ad_{X}}$


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#state/stub | #lang/en | #SemBr