en>David Eppstein: fix ignored accessdate parameter issue

2014-12-07T01:53:22Z

fix ignored accessdate parameter issue

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-02-04T06:50:27Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 08:50, 4 February 2014
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			{{distinguish\|loop (algebra)}}

			In [[mathematics]], '''loop algebras''' are certain types of [[Lie algebra]], of particular interest in [[theoretical physics]].

	~~Last week I woke up and realized - Now~~ [~~http~~://~~www.cinemaudiosociety.org meet and greet~~ with ~~luke bryan~~] ~~I have also been single for some time and following much bullying from friends I now find myself opted for internet dating. They~~ [~~http:~~//~~Www.Bing.com~~/~~search?q=guaranteed&form=MSNNWS&mkt=en-us&pq=guaranteed guaranteed] me that there~~ are ~~a lot~~ of ~~ordinary, pleasant~~ and ~~entertaining folks to fulfill, therefore here goes the message!~~<br>~~My buddies~~ and ~~family~~ are ~~awesome and spending time with them at bar gigabytes or dinners is constantly essential~~. ~~I have never been in~~ to ~~night clubs as I find you could do not get a nice dialogue against~~ the ~~noise. I additionally have two undoubtedly cheeky and really adorable puppies that are constantly excited to meet new individuals.~~<br>~~I try and maintain as physically healthy as potential coming to the gymnasium [http:~~//~~www~~.~~ffpjp24.org luke bryan tours 2014~~] ~~several times a week. I love my athletics and make an effort~~ to ~~perform or see as several~~ a ~~possible~~. ~~Being winter I shall frequently at Hawthorn matches~~. ~~Note~~: ~~In case that you contemplated buying an activity I don'~~t ~~brain, I have experienced the carnage of wrestling fits at [http~~://~~Www~~.~~tumblr.com~~/~~tagged/stocktake stocktake~~] ~~revenue~~.<br><br>~~Here is my web page luke bryan discount code~~ - [~~http://www~~.~~ladyhawkshockey~~.~~org http~~:~~//www.ladyhawkshockey.org/~~],		If <math>\mathfrak{g}</math> is a Lie algebra, the [[tensor product]] of <math>\mathfrak{g}</math> with <math>C^\infty(S^1)</math>,

			:<math>\mathfrak{g}\otimes C^\infty(S^1)</math>,

			the [[associative algebra\|algebra]] of (complex) [[smooth function]]s over the [[n-sphere\|circle]] [[manifold]] S<sup>1</sup> is an infinite-dimensional Lie algebra with the [[Lie bracket of vector fields\|Lie bracket]] given by

			:<math>[g_1\otimes f_1,g_2 \otimes f_2]=[g_1,g_2]\otimes f_1 f_2</math>.

			Here g<sub>1</sub> and g<sub>2</sub> are elements of <math>\mathfrak{g}</math> and f<sub>1</sub> and f<sub>2</sub> are elements of <math>C^\infty(S^1)</math>.

			This isn't precisely what would correspond to the [[direct product]] of infinitely many copies of <math>\mathfrak{g}</math>, one for each point in S<sup>1</sup>, because of the smoothness restriction. Instead, it can be thought of in terms of [[smooth map]] from S<sup>1</sup> to <math>\mathfrak{g}</math>; a smooth parameterized loop in <math>\mathfrak{g}</math>, in other words. This is why it is called the '''loop algebra'''.

			We can take the [[Fourier transform]] on this loop algebra by defining

			:<math>g\otimes t^n</math>

			as

			:<math>g\otimes e^{-in\sigma}</math>

			where

			:0 ≤ σ <2π

			is a coordinatization of S<sup>1</sup>.

			If <math>\mathfrak{g}</math> is a [[semisimple Lie algebra]], then a nontrivial [[Group extension#Central extension\|central extension]] of its loop algebra gives rise to an affine [[Kac-Moody algebra]].

			Similarly, a set of all smooth maps from S<sup>1</sup> to a [[Lie group]] G forms an infinite-dimensional Lie group (Lie group in the sense we can define [[functional derivative]]s over it) called the '''[[loop group]]'''. The Lie algebra of a loop group is the corresponding loop algebra.

			{{DEFAULTSORT:Loop Algebra}}
			[[Category:Lie algebras]]


			{{algebra-stub}}

en>RedBot: r2.7.2) (Robot: Adding es:Factorización con fracciones continuas

2012-07-10T19:56:45Z

r2.7.2) (Robot: Adding es:Factorización con fracciones continuas

New page

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Continued fraction factorization - Revision history

en>David Eppstein: fix ignored accessdate parameter issue

en>Monkbot: Fix CS1 deprecated date parameter errors

en>RedBot: r2.7.2) (Robot: Adding es:Factorización con fracciones continuas