Adjoint representation I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't understand this terminology. For sure, $Ad$ does not enables us to represent elements in the Lie group and $ad$ does not enable us in general to represent every element in $\mathfrak{g}$, so where does the name come from?

The term _representation_ comes from representation theory. In particular, Ad gives us a group representation, and ad gives us a Lie-algebra representation.

In particular: given a Lie group $G$ and $A,B \in G$, $\operatorname{Ad}$ is a map from $G$ to $GL(\mathfrak{g})$ such that $$ \operatorname{Ad}_A \operatorname{Ad}_B = \operatorname{Ad}_{A B} $$ Similarly: given a Lie algebra $\mathfrak g$ and $X,Y \in G$, $\operatorname{ad}$ is a map from $\mathfrak g$ to $\mathfrak{gl(g)}$ such that $$ [\operatorname{ad}_X,\operatorname{ad}_Y] = \operatorname{ad}_{[X,Y]} $$ While these need not be **full** or **faithful** representations, they are representations nevertheless.