Adjoint representation of a Lie algebra

Template:Lie groups Template:Mergeto 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 Template:Mvar of a Lie algebra ${\displaystyle {\mathfrak {g}}}$, one defines the adjoint action of Template:Mvar on ${\displaystyle {\mathfrak {g}}}$ as the map

${\displaystyle \operatorname {ad} _{x}:{\mathfrak {g}}\to {\mathfrak {g}}\qquad {\text{with}}\qquad \operatorname {ad} _{x}(y)=[x,y]}$

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.

Let ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra over a field Template:Mvar. Then the linear mapping

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

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${\displaystyle ({\mathfrak {g}})}$. See below.)

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

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

where ○ denotes composition of linear maps.

If ${\displaystyle {\mathfrak {g}}}$ is finite-dimensional, then End${\displaystyle ({\mathfrak {g}})}$ is isomorphic to ${\displaystyle {\mathfrak {gl}}({\mathfrak {g}})}$, the Lie algebra of the general linear group over the vector space ${\displaystyle {\mathfrak {g}}}$ 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

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

takes the form

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

where Template:Mvar, Template:Mvar, and Template:Mvar are arbitrary elements of ${\displaystyle {\mathfrak {g}}}$.

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 ${\displaystyle {\mathfrak {g}}}$ is a module over itself.

The kernel of ad is, by definition, the center of ${\displaystyle {\mathfrak {g}}}$. Next, we consider the image of ad. Recall that 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 Template:Mvar and Template:Mvar 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 Der${\displaystyle ({\mathfrak {g}})}$, 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

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

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

To be more precise, let Template:Mvar be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψg, with Ψg: GG given by the inner automorphism

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

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

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

where Template:Mvar is the differential and TeG is the tangent space at the origin Template:Mvar (Template:Mvar being the identity element of the group Template:Mvar).

The Lie algebra of Template:Mvar is ${\displaystyle {\mathfrak {g}}}$ = Te G. Since Adg ∈ Aut${\displaystyle ({\mathfrak {g}})}$,   Ad: g ↦ Adg is a map from Template:Mvar 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

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

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector Template:Mvar in the algebra ${\displaystyle {\mathfrak {g}}}$ generates a vector field Template:Mvar in the group Template:Mvar. 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 Template:Mvar considered as a manifold.

Further see the derivative of the exponential map.