- Adjoint endomorphism
In

mathematics , the**adjoint endomorphism**or**adjoint action**is anendomorphism ofLie algebra s that plays a fundamental role in the development of the theory ofLie algebras andLie 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 islinear .**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 thegeneral 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 tomatrix 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 alinear map $delta:mathfrak\{g\}\; ightarrow\; mathfrak\{g\}$ that obeys theLeibniz' 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 constant s of the algebra. That is, let {e^{i}} be a set ofbasis vectors for the algebra, with :$[e^i,e^j]\; =\{c^\{ij\_k\; e^k$. Then the matrix elements for ad_{ei}are given by:$\{left\; [\; extrm\{ad\}\_\{e^i\}\; ight]\; \_k\}^j\; =\; \{c^\{ij\_k$.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 theadjoint representation for aLie 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"_{e}G is thetangent space at the origin "e" ("e" is the identity element of the group "G").The Lie algebra "g" of "G" is "g"="T"

_{e}G. 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_{x}y= ["x","y"] of vectors in $mathfrak\{g\}$ is homomorphic to theLie derivative L_{"X"}"Y" = ["X","Y"] of vector fields on the group "G" considered as amanifold .**References***Fulton-Harris

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