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| In [[mathematics]], a '''Lie group homomorphism''' is a map between [[Lie group]]s
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| :<math>\phi\colon G \to H\,</math>
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| which is both a [[group homomorphism]] and a [[smooth map]]. Lie group homomorphisms are the [[morphism]]s in the [[category (mathematics)|category]] of Lie groups. In the case of complex Lie groups, one naturally requires homomorphisms to be [[holomorphic function]]s. In either the real or complex case, it is actually sufficient to only require the maps to be [[continuous function (topology)|continuous]]. Every continuous homomorphism between Lie groups turns out to be an [[analytic function]].
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| An [[isomorphism]] of Lie groups is a homomorphism whose inverse is also a homomorphism. Equivalently, it is a [[diffeomorphism]] which is also a group homomorphism.
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| Let <math>\phi\colon G \to H</math> be a Lie group homomorphism and let <math>\phi_{*}</math> be its [[Pushforward (differential)|derivative]] at the identity. If we identify the [[Lie algebra]]s of ''G'' and ''H'' with their [[tangent space]]s at the identity then <math>\phi_{*}</math> is a map between the corresponding Lie algebras:
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| :<math>\phi_{*}\colon\mathfrak g \to \mathfrak h</math>
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| One can show that <math>\phi_{*}</math> is actually a [[Lie algebra homomorphism]] (meaning that it is a [[linear map]] which preserves the [[Lie bracket]]). In the language of [[category theory]], we then have a covariant [[functor]] from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.
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| One of the most important properties of Lie group homomorphisms is that the maps <math>\phi</math> and <math>\phi_{*}</math> are related by the [[exponential map]]. For all <math>x\in\mathfrak g</math> we have
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| :<math>\phi(\exp(x)) = \exp(\phi_{*}(x)).\,</math>
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| In other words the following diagram [[commutative diagram|commutes]]:
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| [[File:ExponentialMap-01.png|center]]
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| </div>
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| <ref>http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf</ref>
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| ==Footnotes==
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| {{reflist}}
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| [[Category:Lie groups]]
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| [[Category:Morphisms]]
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Latest revision as of 22:14, 27 November 2014
The author's name is Christy Brookins. Ohio is exactly where my home is but my husband wants us to transfer. Playing badminton is a thing that he is totally addicted to. Distributing manufacturing has been his occupation for some time.
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