Adjoint representation of a Lie algebra
|Group theory → Lie groups
Given an element x of a Lie algebra , one defines the adjoint action of x on as the map with
for all y in .
The concept is closely related to the adjoint representation of a Lie group . In fact, as it turns out, is precisely the differential of at the identity element of the group.
Let be a Lie algebra over a field k. Then the linear mapping
given by is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in . See below.)
Within , the Lie bracket is, by definition, given by the commutator of the two operators:
where denotes composition of linear maps. If is finite-dimensional, then is isomorphic to , the Lie algebra of the general linear group over the vector space and if a basis for it is chosen, the composition corresponds to matrix multiplication.
Using the above definition of the Lie bracket, the Jacobi identity
takes the form
where x, y, and z are arbitrary elements of .
This last identity says that ad really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.
In a more module-theoretic language, the construction simply says that is a module over itself.
for all x and y in the algebra.
That adx is a derivation is a consequence of the Jacobi identity. This implies that the image of under ad is a subalgebra of , the space of all derivations of .
Then the matrix elements for adei are given by
Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
Relation to Ad
To be precise, let G be a Lie group, and let be the mapping with given by the inner automorphism
It is an example of a 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=[x,y] of vectors in is homomorphic to the Lie derivative LXY =[X,Y] of vector fields on the group G considered as a manifold.