Adjoint representation of a Lie algebra

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 ↦ 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.

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 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 .

Structure constants

The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {ei} be a set of basis vectors for the algebra, with

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

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 ↦ Ψg, with Ψg: GG 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 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.

Further see the derivative of the exponential map.


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