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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]].
 
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]]

Revision as of 16:58, 4 March 2014

Prostatic congestion and inflammation can stimulate the prostate urethra inducing frequent spermatorrhea. Despite this amzing incidence, few cases come to light. Coffee drank excessively causes a debilitating effect upon the sexual organs. Studies show that this herb is a natural aphrodisiac which also improves energy and reduces stress levels. For most people, the major change in marriage comes with the birth of the first child.

Not only that, the process of cell repair can also be done well ( the regenerate ) , that old cells replaced with better ones. This environment thus promotes the growth of the anaerobic bacteria. The same high fat diet that narrows arteries and blocks flow to the heart also narrows the arteries that carry blood to the penis. Read effective Herbal Treatment for Male Infertility. Therefore, in order to find out if impotence is inherited or not a research was conducted.

Various medical treatments increase blood flow to the genital region and will allow most men to have an erection. The wide variety of skin deterioration that Pumpkin Peel treats is what tends to make it this sort of an invaluable item amongst other incredible products supplied by Evolve clinic. Wifi enabled laptop sperm (update): Do Wi-Fi enabled laptops kill sperm. The rod is placed in the penis, and the pump is kept at the base of the penis. It is good luck that several of his medicinal secrets are available with some people in India who still practise this method to cure people of ailments.

We keep a large jar of fresh garlic in our refrigerator. However, with new testosterone replacement therapies, many men are beating back these symptoms of low testosterone, with some reporting that they feel and function as well as they did when they were thirty years younger. Do not be alarmed if you or a loved one is occasionally unable to achieve an erection; this is quite normal. Other diseases that can affect the nervous system, such as Alzheimer's disease, multiple sclerosis, and stroke. L-arginine, a well known amino acid is another common cure suggested for reproductive problem.

ED, also known as impotence, can be defined as a total, partial or inconsistent inability to achieve or retain an erection that is satisfactory for the purpose of sexual intercourse. However, now it has been found that in some cases it may be the other way round: These symptoms could manifest as a result of surfing too much pornography. Many persons, therefore, suffer from sexual dysfunctions. It is regular if you are extremely tired or perhaps are under stress or in an troubled state. This medication can benefit a number of mood disorders.

For more information about herbal supplements for erectile dysfunction look into our own web page.

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