Adjoint endomorphism

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Lie groups

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Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group

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Representation theory
Representation of a Lie group Representation of a Lie algebra

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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

ad x (y) = [x, y]

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

ad_x is an action that is linear.

1 Adjoint representation
2 Derivation
3 Structure constants
4 Relation to Ad
5 References

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 [,].

The kernel of $\operatorname{ad}: \mathfrak{g} \to \operatorname{ad}(\mathfrak{g})$ is, by definition, the center of $\mathfrak{g}$ .

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 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

$[e^i,e^j]={c^{ij}}_k e^k.$

Then the matrix elements for ad_eⁱ 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).

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

Ψ g (h) = g h g - 1 .

This is called the Lie group map. Define $Ad g$ to be the derivative of $Ψ g$ at the origin:

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

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 g of G is g=T_eG. Since $\textrm{Ad}_g\in\textrm{Aut}(\mathfrak{g})$ , $\textrm{Ad}:g\mapsto \textrm{Ad}_g$ 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

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

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 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

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

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