Adjoint representation of a Lie algebra

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

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

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

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.

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 ↦ ad_x is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in Der$({\mathfrak {g}})$. See below.)

Within 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 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 ad is, by definition, the center of ${\mathfrak {g}}$. Next, we consider the image of 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 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 more precise, let G be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψ_g, with Ψ_g: G → G given by the inner automorphism

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

It is an example of a Lie group map. Define Ad_g to be the derivative of Ψ_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 being the identity element of the group G).

The Lie algebra of G is ${\mathfrak {g}}$ = T_e G. Since Ad_g ∈ Aut$({\mathfrak {g}})$,   Ad: g ↦ 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) being End(V)).

Then we have

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

The 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.

Further see the derivative of the exponential map.

References

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6.