Adjoint endomorphism


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 mathfrak{g}, one defines the adjoint action of "x" on mathfrak{g} as the endomorphism extrm{ad}_x :mathfrak{g} o mathfrak{g} with

: extrm{ad}_x (y) = [x,y]

for all "y" in mathfrak{g}.

adx is an action that is linear.

Adjoint representation

The mapping extrm{ad}:mathfrak{g} ightarrow extrm{End}(mathfrak{g})=mathfrak{gl}(mathfrak{g}) given by xmapsto extrm{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 extrm{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, : [ extrm{ad}_x, extrm{ad}_y] = extrm{ad}_x circ extrm{ad}_y - extrm{ad}_y circ extrm{ad}_xwhere 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( [ extrm{ad}_x, extrm{ad}_y] ight)(z) = left( extrm{ad}_{ [x,y] } ight)(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} o 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} ightarrow 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 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}.

tructure 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 adeiare given by:{left [ extrm{ad}_{e^i} ight] _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 ightarrow extrm{Aut} (G) be the mapping gmapsto Psi_g with Psi_g:G o G given by the inner automorphism :Psi_g(h)= ghg^{-1}.This is called the Lie group map. Define extrm{Ad}_g to be the derivative of Psi_g at the origin:: extrm{Ad}(g) = (dPsi_g)_e : T_eG ightarrow T_eGwhere "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 extrm{Ad}_gin extrm{Aut}(mathfrak{g}), extrm{Ad}:gmapsto extrm{Ad}_g is a map from "G" to Aut("T"e"G") which will have a derivative from "T"e"G" to End("T"e"G") (the Lie algebra of Aut("V") is End("V")).

Then we have : extrm{ad} = d( extrm{Ad})_e:T_eG ightarrow extrm{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 L"X""Y" = ["X","Y"] of vector fields on the group "G" considered as a manifold.

References

*Fulton-Harris


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