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en>De Riban5: + category

2015-01-08T10:34:56Z

+ category

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188.26.22.131: /* References */ see al

2014-03-04T15:58:44Z

References: see al

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188.26.22.131: micro-macro

2014-01-27T11:24:47Z

micro-macro

← Older revision		Revision as of 13:24, 27 January 2014
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	~~Oscar~~ is ~~how he~~'s called and ~~he totally enjoys this name~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my husband reside~~. ~~My day job~~ is a ~~meter reader~~. ~~One~~ of the ~~things he enjoys most~~ is ~~ice skating but he~~ is ~~having difficulties~~ to ~~find time~~ for it.<br><br>~~Feel free~~ to ~~surf~~ to ~~my weblog~~ ... ~~healthy food delivery~~ (~~[https~~://~~www~~.~~Tumblr~~.~~com~~/~~blog~~/~~mmattchuu sources tell me~~])		{{Lie groups \|Algebras}}

			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]].

			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

			:<math>\operatorname{ad}_x (y) = [x,y]</math>

			for all ''y'' in <math>\mathfrak{g}</math>.

			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.

			== Adjoint representation ==

			Let <math>\mathfrak{g}</math> be a Lie algebra over a field ''k''. Then the [[linear map\|linear mapping]]
			:<math>\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})</math>
			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.)

			Within <math>\operatorname{End}(\mathfrak{g})</math>, the [[Lie bracket]] is, by definition, given by the commutator of the two operators:
			:<math>[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x</math>
			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]].

			Using the above definition of the Lie bracket, the [[Jacobi identity]]
			:<math>[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0</math>
			takes the form
			:<math>\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)</math>
			where ''x'', ''y'', and ''z'' are arbitrary elements of <math>\mathfrak{g}</math>.

			This last identity says that ''ad'' really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.

			In a more module-theoretic language, the construction simply says that <math>\mathfrak{g}</math> is a module over itself.

			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,

			:<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]</math>
			for all ''x'' and ''y'' in the algebra.

			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>.

			== Structure constants ==

			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
			:<math>[e^i,e^j]=\sum_k{c^{ij}}_k e^k.</math>
			Then the matrix elements for
			ad<sub>e<sup>i</sup></sub>
			are given by
			:<math>{\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k. </math>

			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 <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]]
			:<math>\Psi_g(h)= ghg^{-1}.</math>
			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:
			:<math>\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG</math>
			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'').

			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'')).

			Then we have
			:<math>\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).</math>

			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]].

			== References ==
			*{{Fulton-Harris}}

			[[Category:Representation theory of Lie algebras]]
			[[Category:Lie groups]]

en>Protein Chemist: Undid revision 502968915 by 130.15.102.100 (talk)

2012-07-18T21:23:52Z

Undid revision 502968915 by 130.15.102.100 (talk)

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Oscar is how he's called and he totally enjoys this name. South Dakota is exactly where me and my husband reside. My day job is a meter reader. One of the things he enjoys most is ice skating but he is having difficulties to find time for it.<br><br>Feel free to surf to my weblog ... healthy food delivery ([https://www.Tumblr.com/blog/mmattchuu sources tell me])