# 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 with

The concept generates the adjoint representation of a Lie group . In fact, is precisely the differential of at the identity element of the group.

## Adjoint representation

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.

The kernel of is, by definition, the center of . Next, we consider the image of . 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 , 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 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 *T*_{e}G 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(*T*_{e}*G*) which will have a derivative from *T*_{e}*G* to End(*T*_{e}*G*) (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 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.