Adjoint endomorphism

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 <math>\mathfrak{g}<math>, one defines the adjoint action of x on <math>\mathfrak{g}<math> as the endomorphism <math>\textrm{ad}_x :\mathfrak{g}\to \mathfrak{g}<math> with

<math>\textrm{ad}_x (y) = [x,y]<math>

for all y in <math>\mathfrak{g}<math>. Note that ad_x is an action and that it is linear.

Adjoint representation

The mapping <math>\textrm{ad}:\mathfrak{g}\rightarrow \textrm{End}(\mathfrak{g})<math> given by <math>x\mapsto \textrm{ad}_x<math> is a representation of a Lie algebra and is called the adjoint representation of the algebra. Note that physics literature usually uses the notation gl(V) instead of End(V) to denote the set of linear maps of a vector space V (which is the Lie algebra of the general linear group over V); we recall that, of course, <math>\mathfrak{g}<math> is a vector space.

The Jacobi identity

<math>[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0<math>

takes the form

<math>\textrm{ad}_{[x,y]} = [\textrm{ad}_x,\textrm{ad}_y]<math>.

Note that because <math>\textrm{End}(\mathfrak{g})<math> is a set of linear transformations of a vector space, we can take the composition of two maps, and we can then write the Lie bracket as

<math>[\textrm{ad}_x,\textrm{ad}_y]=\textrm{ad}_x \circ \textrm{ad}_y – \textrm{ad}_y \circ \textrm{ad}_x<math>

where <math>\circ<math> denotes composition of linear maps. If a basis is chosen for <math>\mathfrak{g}<math>, this corresponds to matrix multiplication. This last identity allows us to confirm that ad really is a Lie algebra homomorphism, in that the morphism ad commutes with the multiplication operator [,]. To see this, take an element z in g. We then have

<math>\left([\textrm{ad}_x,\textrm{ad}_y]\right)(z)

= [[x,y],z] = \left(\textrm{ad}_{[x,y]}\right)(z)<math>

Derivation

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

<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]<math>

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

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

<math>[e^i,e^j]={c^{ij}}_k e^k<math>.

Then the matrix elements for ad_eⁱ are given by

<math>{\left[ \textrm{ad}_{e^i}\right]_k}^j = {c^{ij}}_k <math>.

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

Relation to Ad

Note that 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 <math>\Psi:G\rightarrow \textrm{Aut} (G)<math> be the mapping <math>g\mapsto \Psi_g<math> with <math>\Psi_g:G\to G<math> given by the inner automorphism

<math>\Psi_g(h)= ghg^{-1}<math>.

This is called the Lie group map. Define <math>\textrm{Ad}_g<math> to be the derivative of <math>\Psi_g<math> at the origin:

<math>\textrm{Ad}(g) = (d\Psi_g)_e : T_eG \rightarrow T_eG<math>

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). Note that the Lie algebra g of G is g=T_eG. Since <math>\textrm{Ad}_g\in\textrm{Aut}(\mathfrak{g})<math>, <math>\textrm{Ad}:g\mapsto \textrm{Ad}_g<math> 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

<math>\textrm{ad} = d(\textrm{Ad})_e:T_eG\rightarrow \textrm{End} (T_eG)<math>.

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

References

• William Fulton and Joe Harris, Representation Theory, A First Course, (1991) Springer-Verlag, New York. ISBN 0–387–97495–4