Adjoint Endomorphism - Relation To Ad

Relation To Ad

Ad and ad are related through the exponential map; crudely, Ad = exp ad, where Ad is the adjoint representation for a Lie group.

To be precise, let G be a Lie group, and let be the mapping with given by the inner automorphism

This is called the Lie group map. Define to be the derivative of at the origin:

where d is the differential and T_eG is the tangent space at the origin e (e is the identity element of the group G).

The Lie algebra of G is . Since, is a map from G to Aut(T_eG) which will have a derivative from T_eG to End(T_eG) (the Lie algebra of Aut(V) is End(V)).

Then we have

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x in the algebra generates a vector field X in the group G. Similarly, the adjoint map ad_xy= of vectors in is homomorphic to the Lie derivative L_XY = of vector fields on the group G considered as a manifold.