Adjoint representation of a Lie algebra

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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[edit]

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 ↦ adx 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 adx 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[edit]

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 adei 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[edit]

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

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

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

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

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 \mathfrak{g} = Te G. Since Adg ∈ Aut(\mathfrak{g}),   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

\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 adxy = [x,y] of vectors in \mathfrak{g} is homomorphic to the Lie derivative LXY = [X,Y] of vector fields on the group G considered as a manifold.

References[edit]