Infinitesimal picture. Differentiating conjugation at the identity yields the adjoint actionAd:G→Aut(g), so conjugation is the global geometric source of the adjoint representation.
A Lie group is a group G equipped with the structure of a smooth manifold such that the group operations are smooth maps:
μ:G×G→G,μ(g,h)=gh,ι:G→G,ι(g)=g−1.
For each g∈G, the left translationLg(h)=gh and the right translationRg(h)=hg are diffeomorphisms of G, with inverses Lg−1 and Rg−1. The tangent space at the identity TeG carries a canonical Lie algebra structure, called the Lie algebra of G, and the exponential map relates this infinitesimal structure to local group behavior near e.
Let G be a Lie group acting smoothly on a manifold M via an action map a:G×M→M (see smooth actions of Lie groups). For a point x∈M, the stabilizer (or isotropy subgroup) at x is
Gx={g∈G∣g⋅x=x}.
It is a subgroup of G. Since it is the preimage of the closed set {x} under the continuous map g↦g⋅x, it is closed in G; hence Gx is a closed subgroup, and by the closed subgroup theorem it is automatically an embedded Lie subgroup.
Lie algebra of the stabilizer
Let g be the Lie algebra of G. For X∈g, let XM denote the fundamental vector field on M generated by X (equivalently, XM(x)=dtdt=0exp(tX)⋅x). Then the Lie algebra of Gx is
is a smooth bijection; under mild hypotheses (e.g. proper actions), it is a diffeomorphism. In the transitive case (see transitive actions) this identifies M with a homogeneous space of the form G/Gx (compare coset spaces).
There is a unique smooth manifold structure on H making it a Lie group such that the inclusion ι:H↪G is a smooth injective immersion and a homeomorphism onto its image. In particular, H is an embedded Lie subgroup of G.
The coset spaceG/H admits a unique smooth manifold structure such that the projection π:G→G/H is a smooth submersion, making G/H into a basic example of a homogeneous space.