Knowlpedia

Adjoint Action of a Lie Group

The conjugation action of a Lie group on itself and the induced linear action on its Lie algebra.

Let GG be a . The conjugation map by g∈Gg\in G is

cg:G→G,cg(h)=ghg−1.c_g:G\to G,\qquad c_g(h)=ghg^{-1}.

This defines a smooth action of GG on itself by group automorphisms.

Induced action on the Lie algebra

Differentiating at the identity gives a linear map

Ad⁡g:=(dcg)e:g→g,g=TeG,\operatorname{Ad}_g := (dc_g)_e : \mathfrak{g} \to \mathfrak{g}, \qquad \mathfrak{g}=T_eG,

called the adjoint action of GG on its Lie algebra. The assignment g↦Ad⁡gg\mapsto \operatorname{Ad}_g is a

Ad⁡:G→Aut⁡(g),\operatorname{Ad}:G\to \operatorname{Aut}(\mathfrak{g}),

where Aut⁡(g)\operatorname{Aut}(\mathfrak{g}) denotes .

Kernel

The kernel of Ad⁡\operatorname{Ad} is the centralizer of the identity component G∘G^\circ. In particular, if GG is connected, then

ker⁡(Ad⁡)=Z(G),\ker(\operatorname{Ad})=Z(G),

the .

Matrix group picture and exponentials

For a matrix Lie group G⊆GL⁡(n)G\subseteq \operatorname{GL}(n),

Ad⁡g(X)=gXg−1.\operatorname{Ad}_g(X)=gXg^{-1}.

Moreover, with the one has the fundamental relation

Ad⁡exp⁡(X)=exp⁡(ad⁡X),\operatorname{Ad}_{\exp(X)} = \exp(\operatorname{ad}_X),

linking Ad⁡\operatorname{Ad} to the and ultimately to the .

Adjoint actions play a central role in structure theory (e.g. , , and the ).