Adjoint endomorphism

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

adx(y) = [x,y]

for all y in \mathfrak{g}.

adx is an action that is linear.

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[edit] Adjoint representation

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

Within \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,

[\textrm{ad}_x,\textrm{ad}_y]=\textrm{ad}_x \circ \textrm{ad}_y - \textrm{ad}_y \circ \textrm{ad}_x

where \circ denotes composition of linear maps. If a basis is chosen for \mathfrak{g}, this corresponds to matrix multiplication.

Using this and the definition of the Lie bracket in terms of the mapping ad above, the Jacobi identity

[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0

takes the form

\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 \mathfrak{g}.

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

[edit] Derivation

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

δ([x,y]) = [δ(x),y] + [x,δ(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 \mathfrak{g} under ad is a subalgebra of \operatorname{Der}(\mathfrak{g}), the space of all derivations of \mathfrak{g}.

[edit] 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]={c^{ij}}_k e^k.

Then the matrix elements for adei are given by

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

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

[edit] 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 \textrm{Aut} (G) be the mapping g\mapsto \Psi_g with \Psi_g:G\to G given by the inner automorphism

Ψg(h) = ghg − 1.

This is called the Lie group map. Define Adg to be the derivative of Ψg at the origin:

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

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 \textrm{Ad}_g\in\textrm{Aut}(\mathfrak{g}), \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

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

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 adxy=[x,y] of vectors in \mathfrak{g} is homomorphic to the Lie derivative LXY =[X,Y] of vector fields on the group G considered as a manifold.

[edit] References

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