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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 ${\mathrm{ad}}_x:\mathfrak{g}\to \mathfrak{g}$ with

${\mathrm{ad}}_x(y)=[x,y]$

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

The concept generates the adjoint representation of a Lie group $\mathrm{Ad}$. In fact, $\mathrm{ad}$ is precisely the differential of $\mathrm{Ad}$ at the identity element of the group.

Adjoint representation

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

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

given by $x\mapsto {\mathrm{ad}}_x$ is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in $\mathrm{Der}(\mathfrak{g})$. See below.)

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

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

where $\circ$ denotes composition of linear maps. If $\mathfrak{g}$ is finite-dimensional, then $\mathrm{End}(\mathfrak{g})$ is isomorphic to $\mathfrak{g}\mathfrak{l}(\mathfrak{g})$, 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

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

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 $\mathrm{ad}$ is, by definition, the center of $\mathfrak{g}$. Next, we consider the image of $\mathrm{ad}$. Recall that a derivation on a Lie algebra is a linear map $\delta :\mathfrak{g}\to \mathfrak{g}$ 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 $\mathrm{Der}(\mathfrak{g})$, 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

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

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 precise, let G be a Lie group, and let $\mathrm{\Psi }:G\to \mathrm{Aut}(G)$ be the mapping $g\mapsto {\mathrm{\Psi }}_g$ with ${\mathrm{\Psi }}_g:G\to G$ given by the inner automorphism

${\mathrm{\Psi }}_g(h)=gh{g}^{-1}.$

It is an example of a Lie group map. Define ${\mathrm{Ad}}_g$ to be the derivative of ${\mathrm{\Psi }}_g$ at the origin:

${\mathrm{Ad}}_g=(d{\mathrm{\Psi }}_g{)}_e:T_{e}G\to T_{e}G$

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

The Lie algebra of G is $\mathfrak{g}=T_{e}G$. Since ${\mathrm{Ad}}_g\in \mathrm{Aut}(\mathfrak{g})$, $\mathrm{Ad}:g\mapsto {\mathrm{Ad}}_g$ 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) is End(V)).

Then we have

$\mathrm{ad}=d(\mathrm{Ad}{)}_e:T_{e}G\to \mathrm{End}(T_{e}G).$

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.

References

• Template:Fulton-Harris