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 $\mathfrak{g}$, one defines the adjoint action of x on $\mathfrak{g}$ as the map

$\operatorname{ad}_x :\mathfrak{g}\to \mathfrak{g} \qquad \text{with} \qquad \operatorname{ad}_x (y) = [x,y]$

for all y in $\mathfrak{g}$.

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 $\mathfrak{g}$ be a Lie algebra over a field k. Then the linear mapping

$\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})$

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$(\mathfrak{g})$. See below.)

Within End$(\mathfrak{g})$, the Lie bracket is, by definition, given by the commutator of the two operators:

$[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x$

where ○ denotes composition of linear maps.

If $\mathfrak{g}$ is finite-dimensional, then End$(\mathfrak{g})$ is isomorphic to $\mathfrak{gl}(\mathfrak{g})$, the Lie algebra of the general linear group over the vector space $\mathfrak{g}$ 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

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$

takes the form

$\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)$

where x, y, and z are arbitrary elements of $\mathfrak{g}$.

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 $\mathfrak{g}$ is a module over itself.

The kernel of ad is, by definition, the center of $\mathfrak{g}$. Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map $\delta:\mathfrak{g}\rightarrow \mathfrak{g}$ that obeys the Leibniz' law, that is,

$\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]$

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 $\mathfrak{g}$ under ad is a subalgebra of Der$(\mathfrak{g})$, the space of all derivations of $\mathfrak{g}$.

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

$[e^i,e^j]=\sum_k{c^{ij}}_k e^k.$

Then the matrix elements for ad_eⁱ are given by

${\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k ~.$

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

$\Psi_g(h)= ghg^{-1}~.$

It is an example of a Lie group map. Define Ad_g to be the derivative of Ψ_g at the origin:

$\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG$

where d is the differential and T_eG is the tangent space at the origin e (e being the identity element of the group G).

The Lie algebra of G is $\mathfrak{g}$ = T_e G. Since Ad_g ∈ Aut$(\mathfrak{g})$,   Ad: g ↦ Ad_g is a map from G to Aut(T_eG) which will have a derivative from T_eG to End(T_eG) (the Lie algebra of Aut(V) being End(V)).

Then we have

$\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).$

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x in the algebra $\mathfrak{g}$ generates a vector field X in the group G. Similarly, the adjoint map ad_xy = [x,y] of vectors in $\mathfrak{g}$ is homomorphic to the Lie derivative L_XY = [X,Y] of vector fields on the group G considered as a manifold.

Further see the derivative of the exponential map.