# Adjoint representation of a Lie algebra

Template:Lie groups
Template:Mergeto
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 Template:Mvar of a Lie algebra , one defines the adjoint action of Template:Mvar on as the map

for all Template:Mvar 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 Template:Mvar. 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 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 Template:Mvar, Template:Mvar, and Template:Mvar 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 Template:Mvar and Template:Mvar 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 Template:Mvar 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 Template:Mvar is the differential and *T*_{e}*G* is the tangent space at the origin Template:Mvar (Template:Mvar being the identity element of the group Template:Mvar).

The Lie algebra of Template:Mvar is = *T*_{e} *G*. Since Ad_{g} ∈ Aut, Ad: *g* ↦ Ad_{g} is a map from Template:Mvar 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 use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector Template:Mvar in the algebra generates a vector field Template:Mvar in the group Template:Mvar. 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 Template:Mvar considered as a manifold.

Further see the derivative of the exponential map.