Adjoint endomorphism

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Template:Lie groups

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 , one defines the adjoint action of x on as the endomorphism with

for all y in .

is an action that is linear.

Adjoint representation

The mapping given by is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Here, is the Lie algebra of the general linear group over the vector space . It is isomorphic to .)

Within , 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,

where denotes composition of linear maps. If is finite-dimensional and a basis for it is chosen, this corresponds to matrix multiplication.

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

takes the form

where x, y, and z are arbitrary elements of .

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 is, by definition, the center of .

Derivation

A derivation on a Lie algebra is a linear map that obeys the Leibniz' law, that is,

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 under ad is a subalgebra of , the space of all derivations of .

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

Then the matrix elements for adei are given by

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 precise, let G be a Lie group, and let be the mapping with given by the inner automorphism

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

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 , 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

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x in the algebra generates a vector field X in the group G. Similarly, the adjoint map adxy=[x,y] of vectors in is homomorphic to the Lie derivative LXY =[X,Y] of vector fields on the group G considered as a manifold.

References

ru:Присоединённое представление алгебры Ли