adjoint representation
Definition
There are two closely related notions that may be mixed up in terminology. By the representation $Ad$ we mean a representation from a lie group to its lie algebra (viewed just as a vector space), where by the representation $ad$ we mean a representation from a lie algebra to itself.
Definition
Let $G$ be a Lie group with lie algebra $\mathfrak{g}$. Then
\begin{gather}
Ad:G\to GL(\mathfrak{g})\
A\mapsto A(-)A^{-1}
\end{gather}
and
\begin{gather}
ad:\mathfrak{g}\to GL(\mathfrak{g}) \
A\mapsto [A,-]
\end{gather}