adjoint representation of Lie group

Definition

The adjoint representation of a Lie group $G$, is the map

$$\begin{array}{rl}\text{Ad}:G& ⟶\text{GL}(\mathfrak{g})\\ g& ⟼{\text{Ad}}_g\end{array}$$

where ${\text{Ad}}_g$ is the map ${\text{Ad}}_g=(d{C}_g{)}_e:T_{e}G\to T_{e}G$ is the differential of conjugation by the element $g\in G$ called the adjoint action.

Intuition

First, given any Lie group, we know that we always have at least one diffeomorphism: conjugation by an element $g\in G$.

$$\begin{array}{rl}{C}_g:G& ⟶G\\ h& ⟼gh{g}^{-1}\end{array}$$

Taking the differential at the identity element gives the map between tangent spaces

$$(d{C}_g{)}_e:T_{e}G⟶T_{ge{g}^{-1}}G.$$

Which we can recognize is

$$(d{C}_g{)}_e:\mathfrak{g}⟶\mathfrak{g}.$$

This ends up being a very special map called the Adjoint action, so we denote $(d{C}_g{)}_e={\text{Ad}}_g$. The Adjoint action shows one way in which the Lie group $G$ acts on its Lie algebra.

Note that $\mathfrak{g}$ is a vector space and ${\text{Ad}}_g\in \text{End}(\mathfrak{g})$, so we can construct the Lie group representation where $\mathfrak{g}$ is the vector space in the representation.

$$\begin{array}{rl}\text{Ad}:G& ⟶\text{GL}(\mathfrak{g})\\ g& ⟼A{d}_g\end{array}$$

This is called the Adjoint representation.