What is the meaning of the adjoint representation ${\rm Ad}:G\times{\frak g}\to{\frak g}$ of a Lie group $G$?

Question

The adjoint representaion of $G$ is a homomorphism $ad_{a}:g \rightarrow aga^{-1}$, $a,g \in G$, what is the meaning of this? Now if we identify $T_{e}G$ with $\mathfrak{g}$ we have the adjoint map $Ad:G \times \mathfrak{g} \rightarrow \mathfrak{g}$, what is the meaning of that? (context of connections (one form) on principal bundles)

Answer 1

$ad$ is a homomorphism $ad : G \mapsto \mathrm{Aut}(G)$. By definition, $ad$ takes $a \in G$ to the inner automorphism $ad_a \in \mathrm{Aut}(G)$ that is defined by the formula $\mathrm{ad}_a(g) = aga^{-1}$.

Notice that for each $a \in G$ we have $\mathrm{ad}_a(e)=aea^{-1}=e$. Therefore the derivative $D_{ad_a} : T_e G \to T_e G$ is defined, it is linear automorphism of $T_e G = \mathfrak{g}$, and it is an automorphism of the Lie algebra structure on $\mathfrak{g}$.

$Ad$ is a homomorphism $Ad : G \mapsto \mathrm{Aut}(\mathfrak{g})$. By definition, $Ad$ takes $a \in G$ to $Ad_a = D_{ad_a}$ which, as seen just above, is a Lie algebra automoprhism of $\mathfrak{g}$.