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| In [[mathematics]], Lie algebroids serve the same role in the theory of [[Lie groupoid]]s that [[Lie algebra]]s serve in the theory of [[Lie groups]]: reducing global problems to infinitesimal ones. Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".
| | True said, The strong bond of love is a driving force behind many socio-economic concepts, but in practical terms, one cannot affix bridges and dams with love, it has to be a strong adhesive, chemically mixed in other important elements. This strong adhesive chemical is known as Epoxy resin.<br><br>There are two main categories of epoxy resins, which are termed as -<br><br>Glycidyl epoxy: to prepare these resins, there is requirement of a condensation reaction<br><br>Non-glycidyl epoxy: to form these epoxies, a preoxidation of olefinic double bond is required<br><br>These two types are further classified in sub categories as:<br><br>Glycidyl epoxy has two classifications such as glycidyl ester and the second one is glycidyl amine.<br><br>Non-glycidyl epoxy has two classifications aliphatic and the second classification is cycloaliphatic.<br><br>In chemical forms, Epoxy resins are defined as a particle, which contains more than one epoxide group, and these groups are known as oxirane or ethoxyline. This scientifically researched chemical is widely used for many purposes such as in the construction, for the purposes of home repair and maintenance and in some instances this is also used by metal working industries.<br><br>However, epoxy based resin is not along competent enough of holding the bond; another catalyzing agent is blended with this resin to harden the resin into a tough adhesive. This particular adhesive has many unmatched properties and those qualities make this adhesive a versatile chemical.<br><br>These thermosetting polymers are basically used for high performance coating, encapsulated material. As these resins are blessed with laudable electrical properties, adhesive adaptability with various metals, free from any worry of shrinkage or moisture; hence they are used widely by marine product manufacturers. If you have any concerns regarding where and how you can make use of [http://prattz1.blog.com/2014/05/29/resistencia-y-implementaciones-distintas-de-la-resina-epoxi/ resina epoxi], you can contact us at our own page. Below are few common uses of Epoxy Resin<br><br>If you are about to paint your walls or windows, use this with your paint and colour protector.<br><br>If your little kiddo broke an expensive piece of China, let his adhesive help you in fixing the same.<br><br>You must have seen jewellery items like necklace, earring, bracelets, but do you know even jewellers use this chemical to fix the joints.<br><br>As we earlier discussed epoxy resin is moisture proof, so mainly boat owners and makers, use this resin to protect their boats (livelihood) from leakage. |
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| More precisely, a '''Lie algebroid'''
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| is a triple <math>(E, [\cdot,\cdot], \rho)</math> consisting of a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>, together with a [[Lie_algebra#Definition_and_first_properties|Lie bracket]] <math>[\cdot,\cdot]</math> on its [[Module (mathematics)|module]] of sections <math>\Gamma (E)</math> and a morphism of vector bundles <math>\rho: E\rightarrow TM</math> called the '''anchor'''. Here <math>TM</math> is the [[tangent bundle]] of <math>M</math>. The anchor and the bracket are to satisfy the Leibniz rule:
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| :<math>[X,fY]=\rho(X)f\cdot Y + f[X,Y]</math> | |
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| where <math>X,Y \in \Gamma(E), f\in C^\infty(M)</math> and <math>\rho(X)f</math> is the [[derivative]] of <math>f</math> along the vector field <math>\rho(X)</math>. It follows that
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| :<math>\rho([X,Y])=[\rho(X),\rho(Y)] </math>
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| for all <math>X,Y \in \Gamma(E)</math>.
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| == Examples ==
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| * Every [[Lie algebra]] is a Lie algebroid over the one point manifold.
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| * The tangent bundle <math>TM</math> of a manifold <math>M</math> is a Lie algebroid for the [[Lie bracket of vector fields]] and the identity of <math>TM</math> as an anchor.
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| * Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
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| * Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
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| * To every [[Lie groupoid]] is associated a Lie algebroid, generalizing how a Lie algebra is associated to a [[Lie group]] (see also below). For example, the Lie algebroid <math>TM</math> comes from the pair groupoid whose objects are <math>M</math>, with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible,<ref>Marius Crainic, Rui L. Fernandes. [http://arxiv.org/abs/math/0105033 Integrability of Lie brackets], Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620</ref> but every Lie algebroid gives a [[Algebraic stack|stacky]] Lie groupoid.<ref>Hsian-Hua Tseng and Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, available as [http://arxiv.org/abs/math/0405003 arXiv:math/0405003]</ref><ref>Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, available as [http://arxiv.org/abs/math/0701024 arXiv:math/0701024]</ref>
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| * Given the action of a Lie algebra g on a manifold M, the set of g -invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
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| * The [[Atiyah algebroid]] of a [[principal bundle|principal ''G''-bundle]] ''P'' over a manifold ''M'' is a Lie algebroid with [[short exact sequence]]:
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| *:<math> 0 \to P\times_G \mathfrak g\to TP/G\xrightarrow{\rho} TM \to 0.</math>
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| : The space of sections of the Atiyah algebroid is the Lie algebra of ''G''-invariant vector fields on ''P''.
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| == Lie algebroid associated to a Lie groupoid ==
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| To describe the construction let us fix some notation. ''G'' is the space of morphisms of the Lie groupoid, ''M'' the space of objects, <math>e:M\to G</math> the units and <math>t:G\to M</math> the target map.
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| <math>T^tG=\bigcup_{p\in M}T(t^{-1}(p))\subset TG</math> the ''t''-fiber tangent space. The Lie algebroid is now the vector bundle <math>A:=e^*T^tG</math>. This inherits a bracket from ''G'', because we can identify the ''M''-sections into ''A'' with left-invariant vector fields on ''G''. The anchor map then is obtained as the derivation of the source map
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| <math>Ts:e^*T^tG \rightarrow TM </math>. Further these sections act on the smooth functions of ''M'' by identifying these with left-invariant functions on ''G''.
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| As a more explicit example consider the Lie algebroid associated to the pair groupoid <math>G:=M\times M</math>. The target map is <math>t:G\to M: (p,q)\mapsto p</math> and the units <math>e:M\to G: p\mapsto (p,p)</math>. The ''t''-fibers are <math>p\times M</math> and therefore <math>T^tG=\bigcup_{p\in M}p\times TM \subset TM\times TM</math>. So the Lie algebroid is the vector bundle <math>A:=e^*T^tG=\bigcup_{p\in M} T_pM=TM</math>. The extension of sections ''X'' into ''A'' to left-invariant vector fields on ''G'' is simply <math>\tilde X(p,q)=0\oplus X(q)</math> and the extension of a smooth function ''f'' from ''M'' to a left-invariant function on ''G'' is <math>\tilde f(p,q)=f(q)</math>. Therefore the bracket on ''A'' is just the Lie bracket of tangent vector fields and the anchor map is just the identity.
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| Of course you could do an analog construction with the source map and right-invariant vector fields/ functions. However you get an isomorphic Lie algebroid, with the explicit isomorphism <math>i_*</math>, where <math>i:G\to G</math> is the inverse map.
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| ==See also==
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| *[[R-algebroid]]
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| ==References==
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| <references/>
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| ==External links==
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| *Alan Weinstein, Groupoids: unifying internal and external
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| symmetry, ''AMS Notices'', '''43''' (1996), 744-752. Also available as [http://arxiv.org/abs/math/9602220 arXiv:math/9602220]
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| *Kirill Mackenzie, ''Lie Groupoids and Lie Algebroids in Differential Geometry'', Cambridge U. Press, 1987.
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| *Kirill Mackenzie, ''General Theory of Lie Groupoids and Lie Algebroids'', Cambridge U. Press, 2005
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| *Charles-Michel Marle, ''Differential calculus on a Lie algebroid and Poisson manifolds'' (2002). Also available in [http://arxiv.org/abs/0804.2451v1 arXiv:0804.2451]
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| {{DEFAULTSORT:Lie Algebroid}}
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| [[Category:Lie algebras]]
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| [[Category:Differential geometry]]
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True said, The strong bond of love is a driving force behind many socio-economic concepts, but in practical terms, one cannot affix bridges and dams with love, it has to be a strong adhesive, chemically mixed in other important elements. This strong adhesive chemical is known as Epoxy resin.
There are two main categories of epoxy resins, which are termed as -
Glycidyl epoxy: to prepare these resins, there is requirement of a condensation reaction
Non-glycidyl epoxy: to form these epoxies, a preoxidation of olefinic double bond is required
These two types are further classified in sub categories as:
Glycidyl epoxy has two classifications such as glycidyl ester and the second one is glycidyl amine.
Non-glycidyl epoxy has two classifications aliphatic and the second classification is cycloaliphatic.
In chemical forms, Epoxy resins are defined as a particle, which contains more than one epoxide group, and these groups are known as oxirane or ethoxyline. This scientifically researched chemical is widely used for many purposes such as in the construction, for the purposes of home repair and maintenance and in some instances this is also used by metal working industries.
However, epoxy based resin is not along competent enough of holding the bond; another catalyzing agent is blended with this resin to harden the resin into a tough adhesive. This particular adhesive has many unmatched properties and those qualities make this adhesive a versatile chemical.
These thermosetting polymers are basically used for high performance coating, encapsulated material. As these resins are blessed with laudable electrical properties, adhesive adaptability with various metals, free from any worry of shrinkage or moisture; hence they are used widely by marine product manufacturers. If you have any concerns regarding where and how you can make use of resina epoxi, you can contact us at our own page. Below are few common uses of Epoxy Resin
If you are about to paint your walls or windows, use this with your paint and colour protector.
If your little kiddo broke an expensive piece of China, let his adhesive help you in fixing the same.
You must have seen jewellery items like necklace, earring, bracelets, but do you know even jewellers use this chemical to fix the joints.
As we earlier discussed epoxy resin is moisture proof, so mainly boat owners and makers, use this resin to protect their boats (livelihood) from leakage.