Zoran Skoda adjoint action

UNDER CONSTRUCTION

The conjugation action of a group on itself is another name of the action by the corresponding inner automorphism $(g,h)\mapsto g h g^{-1}$. It is also sometimes called adjoint action.

A Lie algebra $L$ acts on itself by the commutator $ad a (b) = [a,b]$; $ad : L \to End(L)$ is the adjoint representation of the Lie algebra.

A Hopf algebra acts on itself by an adjoint action $(g,h)\mapsto \sum g_{(1)}h S(g_{(2)})$ (where we used Sweedler notation and $S$ is the antipode map).

The adjoint action of a Lie group $G$ with unit element $e$ us the action $Ad : G\to Aut(\mathfrak{g})$ on its own Lie algebra $\mathfrak{g} = T_e G$ is given by the derivative (the tangent map) $Ad_g := d (L_g R_{g^{-1}})_e : \mathfrak{g} = T_e G \to T_{g e g^{-1}} G = T_e G$ of the conjugation action $(g,x)\mapsto g x g^{-1}$ at $e$. If $X = \frac{d \gamma}{d t} |_{t = 0}\in \mathfrak{g}$ for some curve $\gamma(t) = exp(t X)$ around $e$, then …

The adjoint action of the Lie algebra on itself is the differential of the Lie group action

$$ad(X)(Y) = \frac{d}{d t} (Ad(exp(t X)) Y) |_{t = 0} = [X, Y]$$

• wikipedia adjoint representation, Baker-Campbell-Hausdorff formula

For the matrix groups, action of $G$ on $T_e G$ is given by the matrix formula $Ad(g) (x) = g x g^{-1}$.

The Killing form on $\mathfrak{g}$ is the invariant bilinear form $(X,Y)\mapsto Tr(ad(X) ad(Y))$.

Hadamard’s formula

$$exp(X) Y exp(-X) = \sum_{n= 0}^\infty \frac{(ad X)^n Y}{n!}$$

is often useful. The right hand side can be written symbolically as $exp(ad(X))Y$.