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 TeG 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(TeG) which will have a derivative from TeG to End(TeG) (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 adxy= of vectors in is homomorphic to the Lie derivative LXY = of vector fields on the group G considered as a manifold.
Read more about this topic: Adjoint Endomorphism
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