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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?

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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.

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The linear map $\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})$, given by $x ↦ ad(x)$ is a representation of a Lie algebra, so it does enable us to "represent every element in $\mathfrak{g}$" by matrices, though not always faithfully (ad is faithful if and only if the Lie algebra has a trivial center). For example, let $\mathfrak{g}=\mathfrak{sl}(2,K)$ be the $3$-dimensional Lie algebra consisting of traceless $2\times2$-matrices. Then $ad$ represents every element $x$ of $\mathfrak{g}$ faithfully by the matrix $ad(x)$. For details see here.

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