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.

Given an element x of a Lie algebra ${\displaystyle {\mathfrak {g}}}$, one defines the adjoint action of x on ${\displaystyle {\mathfrak {g}}}$ as the endomorphism ${\displaystyle {\textrm {ad}}_{x}:{\mathfrak {g}}\to {\mathfrak {g}}}$ with

${\displaystyle {\textrm {ad}}_{x}(y)=[x,y]}$

The mapping ${\displaystyle {\textrm {ad}}:{\mathfrak {g}}\rightarrow \operatorname {Der} ({\mathfrak {g}})\subset {\textrm {End}}({\mathfrak {g}})={\mathfrak {gl}}({\mathfrak {g}})}$ given by ${\displaystyle x\mapsto {\textrm {ad}}_{x}}$ is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Here, ${\displaystyle {\mathfrak {gl}}({\mathfrak {g}})}$ is the Lie algebra of the general linear group over the vector space ${\displaystyle {\mathfrak {g}}}$. It is isomorphic to ${\displaystyle {\textrm {End}}({\mathfrak {g}})}$.)

Within ${\displaystyle {\mathfrak {gl}}({\mathfrak {g}})}$, 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,

${\displaystyle [{\textrm {ad}}_{x},{\textrm {ad}}_{y}]={\textrm {ad}}_{x}\circ {\textrm {ad}}_{y}-{\textrm {ad}}_{y}\circ {\textrm {ad}}_{x}}$

where ${\displaystyle \circ }$ denotes composition of linear maps. If ${\displaystyle {\mathfrak {g}}}$ 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

${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}$

takes the form

${\displaystyle \left([{\textrm {ad}}_{x},{\textrm {ad}}_{y}]\right)(z)=\left({\textrm {ad}}_{[x,y]}\right)(z)}$

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

This last identity confirms that ad really is a Lie algebra homomorphism, in that the morphism ad commutes with the multiplication operator [,].

## Derivation

A derivation on a Lie algebra is a linear map ${\displaystyle \delta :{\mathfrak {g}}\rightarrow {\mathfrak {g}}}$ that obeys the Leibniz' law, that is,

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

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

## 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

${\displaystyle [e^{i},e^{j}]=\sum _{k}{c^{ij}}_{k}e^{k}.}$

Then the matrix elements for adei are given by

${\displaystyle {\left[{\textrm {ad}}_{e^{i}}\right]_{k}}^{j}={c^{ij}}_{k}.}$

Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).

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

${\displaystyle \Psi _{g}(h)=ghg^{-1}.}$

This is called the Lie group map. Define ${\displaystyle {\textrm {Ad}}_{g}}$ to be the derivative of ${\displaystyle \Psi _{g}}$ at the origin:

${\displaystyle {\textrm {Ad}}(g)=(d\Psi _{g})_{e}:T_{e}G\rightarrow T_{e}G}$

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 ${\displaystyle {\textrm {Ad}}_{g}\in {\textrm {Aut}}({\mathfrak {g}})}$, ${\displaystyle {\textrm {Ad}}:g\mapsto {\textrm {Ad}}_{g}}$ 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

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

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