Adjoint representation of a Lie algebra
Group theory → Lie groups Lie groups 


In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras.
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.
Adjoint representation
Let be a Lie algebra over a field k. Then the linear mapping
given by x ↦ ad_{x} 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 finitedimensional, 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 moduletheoretic language, the construction simply says that is a module over itself.
The kernel of ad is, by definition, the center of . Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map that obeys the Leibniz' law, that is,
for all x and y in the algebra.
That ad_{x} 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 .
Structure constants
The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {e^{i}} be a set of basis vectors for the algebra, with
Then the matrix elements for ad_{ei} are given by
Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
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 more precise, let G be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψ_{g}, with Ψ_{g}: G → G given by the inner automorphism
It is an example of a Lie group map. Define Ad_{g} to be the derivative of Ψ_{g} at the origin:
where d is the differential and T_{e}G is the tangent space at the origin e (e being the identity element of the group G).
The Lie algebra of G is = T_{e} G. Since Ad_{g} ∈ Aut, Ad: g ↦ Ad_{g} is a map from G to Aut(T_{e}G) which will have a derivative from T_{e}G to End(T_{e}G) (the Lie algebra of Aut(V) being End(V)).
Then we have
The uppercase/lowercase 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_{x}y = [x,y] of vectors in is homomorphic to the Lie derivative L_{X}Y = [X,Y] of vector fields on the group G considered as a manifold.
Further see the derivative of the exponential map.
References
 Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: SpringerVerlag. ISBN 9780387974958. MR 1153249, ISBN 9780387975276.