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
for all y in .
The concept generates the adjoint representation of a Lie group Ad. In fact, ad is the differential of Ad at the identity element of the group.
Let be a Lie algebra over a field k. Then the linear mapping
given by x ↦ adx is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in Der. See below.)
Within End, 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 End 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 Der, 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 more precise, let G be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψg, with Ψg: G → G given by the inner automorphism
It is an example of a Lie group map. Define Adg to be the derivative of Ψg at the origin:
where d is the differential and TeG is the tangent space at the origin e (e being the identity element of the group G).
The Lie algebra of G is = Te G. Since Adg ∈ Aut, Ad: g ↦ Adg is a map from G to Aut(TeG) which will have a derivative from TeG to End(TeG) (the Lie algebra of Aut(V) being 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.