Zoran Skoda
adjoint action


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

A Lie algebra LL acts on itself by the commutator ada(b)=[a,b]ad a (b) = [a,b]; ad:LEnd(L)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)g (1)hS(g (2))(g,h)\mapsto \sum g_{(1)}h S(g_{(2)}) (where we used Sweedler notation and SS is the antipode map).

The adjoint action of a Lie group GG with unit element ee us the action Ad:GAut(𝔤)Ad : G\to Aut(\mathfrak{g}) on its own Lie algebra 𝔤=T eG\mathfrak{g} = T_e G is given by the derivative (the tangent map) Ad g:=d(L gR g 1) e:𝔤=T eGT geg 1G=T eGAd_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)gxg 1(g,x)\mapsto g x g^{-1} at ee. If X=dγdt| t=0𝔤X = \frac{d \gamma}{d t} |_{t = 0}\in \mathfrak{g} for some curve γ(t)=exp(tX)\gamma(t) = exp(t X) around ee, then …

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

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

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

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

Hadamard’s formula

exp(X)Yexp(X)= n=0 (adX) nYn! 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))Yexp(ad(X))Y.

Created on August 18, 2011 at 00:59:47. See the history of this page for a list of all contributions to it.