Britney Gallivan - Revision history

2.110.114.86: /* Paper folding theorem */ change layout of vaiables for consistency

2014-11-27T20:14:51Z

Paper folding theorem: change layout of vaiables for consistency

← Older revision		Revision as of 22:14, 27 November 2014
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	~~<div style="background:white;">~~		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.<br><br>Feel free to surf to my webpage certified psychics ([http://kard.dk/?p=24252 kard.dk])

	~~In [[mathematics]], a '''Lie group homomorphism''~~' ~~is a map between [[Lie group]]~~s
	~~:<math>\phi\colon G \to H\,</math>~~
	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]].

	~~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.~~

	~~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:~~
	~~:<math>\phi_{*}\colon\mathfrak g \to \mathfrak h</math>~~
	~~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~~.

	~~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~~
	~~:<math>\phi(\exp~~(~~x)) = \exp(\phi_{*}(x)).\,</math>~~
	~~In other words the following diagram~~ [~~[commutative diagram\|commutes]]:~~
	~~[[File:ExponentialMap-01.png\|center]]~~

	~~</div>~~
	~~<ref>~~http://~~www~~.~~math.sunysb.edu~~/~~~vkiritch/MAT552/ProblemSet1~~.~~pdf</ref>~~
	~~==Footnotes==~~
	~~{{reflist}}~~

	~~[[Category:Lie groups]]~~
	~~[[Category:Morphisms]~~]

90.204.47.8: /* Biography */

2013-12-12T20:55:55Z

Biography

← Older revision		Revision as of 22:55, 12 December 2013
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	~~Jayson Berryhill~~ is ~~how I'm called~~ and ~~my spouse doesn't like~~ it at ~~all~~. It's ~~not~~ a ~~common thing but what I like doing~~ is to ~~climb but I don't have~~ the ~~time recently~~. ~~Invoicing~~ is ~~my occupation~~. ~~Her family lives~~ in ~~Ohio.~~<br><br>~~Here is my web site~~ ..~~. [~~http://~~breenq~~.~~com/index~~.~~php?do=~~/~~profile-1144~~/~~info~~/ ~~online psychics~~]		<div style="background:white;">

			In [[mathematics]], a '''Lie group homomorphism''' is a map between [[Lie group]]s
			:<math>\phi\colon G \to H\,</math>
			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]].

			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.

			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:
			:<math>\phi_{*}\colon\mathfrak g \to \mathfrak h</math>
			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.

			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
			:<math>\phi(\exp(x)) = \exp(\phi_{*}(x)).\,</math>
			In other words the following diagram [[commutative diagram\|commutes]]:
			[[File:ExponentialMap-01.png\|center]]

			</div>
			<ref>http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf</ref>
			==Footnotes==
			{{reflist}}

			[[Category:Lie groups]]
			[[Category:Morphisms]]

101.51.6.40 at 07:10, 18 August 2012

2012-08-18T07:10:31Z

New page

Jayson Berryhill is how I'm called and my spouse doesn't like it at all. It's not a common thing but what I like doing is to climb but I don't have the time recently. Invoicing is my occupation. Her family lives in Ohio.<br><br>Here is my web site ... [http://breenq.com/index.php?do=/profile-1144/info/ online psychics]