adjoint representation of Lie group

Definition

The adjoint representation of a Lie group , is the map

where is the map is the differential of conjugation by the element called the adjoint action.

Intuition

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

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

Which we can recognize is

This ends up being a very special map called the Adjoint action, so we denote . The Adjoint action shows one way in which the Lie group acts on its Lie algebra.

Note that is a vector space and , so we can construct the Lie group representation where is the vector space in the representation.

This is called the Adjoint representation.