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| {{Lie groups |Algebras}}
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| In [[mathematics]], the '''adjoint endomorphism''' or '''adjoint action''' is a [[homomorphism]] of [[Lie algebra]]s that plays a fundamental role in the development of the theory of [[Lie algebras]].
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| Given an element ''x'' of a Lie algebra <math>\mathfrak{g}</math>, one defines the adjoint action of ''x'' on <math>\mathfrak{g}</math> as the map <math>\operatorname{ad}_x :\mathfrak{g}\to \mathfrak{g}</math> with
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| :<math>\operatorname{ad}_x (y) = [x,y]</math>
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| for all ''y'' in <math>\mathfrak{g}</math>.
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| The concept generates the [[adjoint representation of a Lie group]] <math>\operatorname{Ad}</math>. In fact, <math>\operatorname{ad}</math> is precisely the differential of <math>\operatorname{Ad}</math> at the identity element of the group.
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| == Adjoint representation ==
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| Let <math>\mathfrak{g}</math> be a Lie algebra over a field ''k''. Then the [[linear map|linear mapping]]
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| :<math>\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})</math>
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| given by <math>x\mapsto \operatorname{ad}_x</math> is a [[representation of a Lie algebra]] and is called the '''adjoint representation''' of the algebra. (Its image actually lies in <math>\operatorname{Der}(\mathfrak{g})</math>. See below.)
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| Within <math>\operatorname{End}(\mathfrak{g})</math>, the [[Lie bracket]] is, by definition, given by the commutator of the two operators:
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| :<math>[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x</math>
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| where <math>\circ</math> denotes composition of linear maps. If <math>\mathfrak{g}</math> is finite-dimensional, then <math>\operatorname{End}(\mathfrak{g})</math> is isomorphic to <math>\mathfrak{gl}(\mathfrak{g})</math>, the Lie algebra of the [[general linear group]] over the vector space <math>\mathfrak{g}</math> and if a basis for it is chosen, the composition corresponds to [[matrix multiplication]].
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| Using the above definition of the Lie bracket, the [[Jacobi identity]]
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| :<math>[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0</math>
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| takes the form
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| :<math>\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)</math>
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| where ''x'', ''y'', and ''z'' are arbitrary elements of <math>\mathfrak{g}</math>.
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| This last identity says that ''ad'' really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.
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| In a more module-theoretic language, the construction simply says that <math>\mathfrak{g}</math> is a module over itself.
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| The kernel of <math>\operatorname{ad}</math> is, by definition, the center of <math>\mathfrak{g}</math>. Next, we consider the image of <math>\operatorname{ad}</math>. Recall that a '''[[derivation (abstract algebra)|derivation]]''' on a Lie algebra is a [[linear map]] <math>\delta:\mathfrak{g}\rightarrow \mathfrak{g}</math> that obeys the [[General Leibniz rule|Leibniz' law]], that is,
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| :<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]</math>
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| for all ''x'' and ''y'' in the algebra.
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| That ad<sub>x</sub> is a derivation is a consequence of the Jacobi identity. This implies that the image of <math>\mathfrak{g}</math> under ''ad'' is a subalgebra of <math>\operatorname{Der}(\mathfrak{g})</math>, the space of all derivations of <math>\mathfrak{g}</math>.
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| == Structure constants ==
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| The explicit matrix elements of the adjoint representation are given by the [[structure constants]] of the algebra. That is, let {e<sup>i</sup>} be a set of [[basis vectors]] for the algebra, with
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| :<math>[e^i,e^j]=\sum_k{c^{ij}}_k e^k.</math>
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| Then the matrix elements for
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| ad<sub>e<sup>i</sup></sub>
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| are given by
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| :<math>{\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k. </math>
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| Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
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| == Relation to Ad ==
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| Ad and ad are related through the [[exponential map]]; crudely, Ad = exp ad, where Ad is the [[adjoint representation]] for a [[Lie group]].
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| To be precise, let ''G'' be a Lie group, and let <math>\Psi:G\rightarrow \operatorname{Aut} (G)</math> be the mapping <math>g\mapsto \Psi_g</math> with <math>\Psi_g:G\to G</math> given by the [[inner automorphism]]
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| :<math>\Psi_g(h)= ghg^{-1}.</math>
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| It is an example of a Lie group map. Define <math>\operatorname{Ad}_g</math> to be the [[tangent space|derivative]] of <math>\Psi_g</math> at the origin:
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| :<math>\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG</math>
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| where ''d'' is the differential and ''T''<sub>e</sub>G is the [[tangent space]] at the origin ''e'' (''e'' is the identity element of the group ''G'').
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| The Lie algebra of ''G'' is <math>\mathfrak{g} = T_e G</math>. Since <math>\operatorname{Ad}_g\in\operatorname{Aut}(\mathfrak{g})</math>, <math>\operatorname{Ad}:g\mapsto \operatorname{Ad}_g</math> is a map from ''G'' to Aut(''T''<sub>e</sub>''G'') which will have a derivative from ''T''<sub>e</sub>''G'' to End(''T''<sub>e</sub>''G'') (the Lie algebra of Aut(''V'') is End(''V'')).
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| Then we have
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| :<math>\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).</math>
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| The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector ''x'' in the algebra <math>\mathfrak{g}</math> generates a [[vector field]] ''X'' in the group ''G''. Similarly, the adjoint map ad<sub>x</sub>y=[''x'',''y''] of vectors in <math>\mathfrak{g}</math> is homomorphic to the [[Lie derivative]] L<sub>''X''</sub>''Y'' =[''X'',''Y''] of vector fields on the group ''G'' considered as a [[manifold]].
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| == References ==
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| *{{Fulton-Harris}}
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| [[Category:Representation theory of Lie algebras]]
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| [[Category:Lie groups]]
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The writer is known as Irwin. Years ago we moved to North Dakota. Doing ceramics is what my family members and I enjoy. I am a meter reader but I strategy on changing it.
Here is my website diet meal delivery