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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 $\operatorname {ad} _{x}:{\mathfrak {g}}\to {\mathfrak {g}}$ with

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

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

The concept generates the adjoint representation of a Lie group $\operatorname {Ad}$. In fact, $\operatorname {ad}$ is precisely the differential of $\operatorname {Ad}$ 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 $x\mapsto \operatorname {ad} _{x}$ is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in $\operatorname {Der} ({\mathfrak {g}})$. See below.)

Within $\operatorname {End} ({\mathfrak {g}})$, 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 $\circ$ denotes composition of linear maps. If ${\mathfrak {g}}$ is finite-dimensional, then $\operatorname {End} ({\mathfrak {g}})$ 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 $\operatorname {ad}$ is, by definition, the center of ${\mathfrak {g}}$. Next, we consider the image of $\operatorname {ad}$. 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 $\operatorname {Der} ({\mathfrak {g}})$, 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 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 precise, let G be a Lie group, and let $\Psi :G\rightarrow \operatorname {Aut} (G)$ be the mapping $g\mapsto \Psi _{g}$ with $\Psi _{g}:G\to G$ given by the inner automorphism

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

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

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

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

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

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