In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups.

${\textrm {ad}}_{x}(y)=[x,y]$ Within ${\mathfrak {gl}}({\mathfrak {g}})$ , the composition of two maps is well defined, and the Lie bracket may be shown to be given by the commutator of the two elements,

$[{\textrm {ad}}_{x},{\textrm {ad}}_{y}]={\textrm {ad}}_{x}\circ {\textrm {ad}}_{y}-{\textrm {ad}}_{y}\circ {\textrm {ad}}_{x}$ where $\circ$ denotes composition of linear maps. If ${\mathfrak {g}}$ is finite-dimensional and a basis for it is chosen, this corresponds to matrix multiplication.

Using this and the definition of the Lie bracket in terms of the mapping ad above, the Jacobi identity

$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ takes the form

$\left([{\textrm {ad}}_{x},{\textrm {ad}}_{y}]\right)(z)=\left({\textrm {ad}}_{[x,y]}\right)(z)$ This last identity confirms that ad really is a Lie algebra homomorphism, in that the morphism ad commutes with the multiplication operator [,].

Derivation

$\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$ 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 $\operatorname {Der} ({\mathfrak {g}})$ , 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 adei are given by

${\left[{\textrm {ad}}_{e^{i}}\right]_{k}}^{j}={c^{ij}}_{k}.$ Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).

Ad and ad are related through the exponential map; crudely, Ad = exp ad, where Ad is the adjoint representation for a Lie group.

$\Psi _{g}(h)=ghg^{-1}.$ ${\textrm {Ad}}(g)=(d\Psi _{g})_{e}:T_{e}G\rightarrow T_{e}G$ where d is the differential and TeG is the tangent space at the origin e (e is the identity element of the group G).

The Lie algebra g of G is g=TeG. Since ${\textrm {Ad}}_{g}\in {\textrm {Aut}}({\mathfrak {g}})$ , ${\textrm {Ad}}:g\mapsto {\textrm {Ad}}_{g}$ is a map from G to Aut(TeG) which will have a derivative from TeG to End(TeG) (the Lie algebra of Aut(V) is End(V)).

Then we have

${\textrm {ad}}=d({\textrm {Ad}})_{e}:T_{e}G\rightarrow {\textrm {End}}(T_{e}G).$ 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 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.