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 $mathfrak\{g\}$.

ad_x 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, :$[\; extrm\{ad\}\_x,\; extrm\{ad\}\_y]\; =\; extrm\{ad\}\_x\; circ\; extrm\{ad\}\_y\; -\; extrm\{ad\}\_y\; circ\; extrm\{ad\}\_x$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 :$left(\; [\; extrm\{ad\}\_x,\; extrm\{ad\}\_y]\; ight)(z)\; =\; left(\; extrm\{ad\}\_\{\; [x,y]\; \}\; ight)(z)$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 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 $mathfrak\{g\}$.

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 ad_eⁱare 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::$extrm\{Ad\}(g)\; =\; (dPsi\_g)\_e\; :\; T\_eG\; ightarrow\; T\_eG$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 $extrm\{Ad\}\_gin\; extrm\{Aut\}(mathfrak\{g\})$, $extrm\{Ad\}:gmapsto\; extrm\{Ad\}\_g$ 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 ad_xy= ["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