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{{Group theory sidebar |Topological}}
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{{Lie groups |Other}}
{{for|groups of actors involved in re-recording movie dialogue during post-production (commonly known in the entertainment industry as "loop groups")|Dubbing (filmmaking)}}
 
In [[mathematics]], a '''loop group''' is a [[Group (mathematics)|group]] of [[Loop (topology)|loops]] in a [[topological group]] ''G'' with multiplication defined pointwise. Specifically, let ''LG'' denote the [[Topological space|space]] of [[Continuous function (topology)|continuous maps]]<math>S^1 \to G</math> equipped with the [[compact-open topology]]. An element of <math>LG</math> is called a ''loop'' in G.  Pointwise multiplication of such loops gives <math>LG</math> the structure of a topological group. The space <math>LG</math> is called the '''free loop group''' on <math>G</math>. A loop group is any [[subgroup]] of the free loop group <math>LG</math>.
 
An important example of a loop group is the group
:<math>\Omega G \,</math>
 
of based loops on <math>G</math>. It is defined to be the kernel of the evaluation map
:<math>e_1: LG \to G</math>,
 
and hence is a closed normal subgroup of <math>LG</math>. (Here, <math>e_1</math> is the map that sends a loop to its value at <math>1</math>.) Note that we may embed <math>G</math> into <math>LG</math> as the subgroup of constant loops.  Consequently, we arrive at a split exact sequence
:<math>1\to \Omega G \to LG \to G\to 1</math>.
 
The space <math>LG</math> splits as a [[semi-direct product]],
:<math>LG = \Omega G \rtimes G</math>.
 
We may also think of <math>\Omega G</math> as the [[loop space]] on <math>G</math>. From this point of view, <math>\Omega G</math> is an [[H-space|''H''-space]] with respect to concatenation of loops.  On the face of it, this seems to provide <math>\Omega G</math> with two very different product maps.  However, it can be shown that concatenation and pointwise multiplication are [[homotopy|homotopic]]. Thus, in terms of the homotopy theory of <math>\Omega G</math>, these maps are interchangeable.
 
Loop groups were used to explain the phenomenon of [[Bäcklund transform]]s in [[soliton]] equations by [[Chuu-Lian Terng]] and [[Karen Uhlenbeck]].<ref>[http://www.ams.org/notices/200001/fea-terng.pdf Geometry of Solitons] by Chuu-Lian Terng and Karen Uhlenbeck</ref>
 
== Notes ==
{{reflist}}
 
== References ==
*{{citation |mr=0900587 |last=Pressley |first=Andrew |last2=Segal |first2=Graeme |title=Loop groups |series=Oxford Mathematical Monographs. Oxford Science Publications |publisher=The Clarendon Press, Oxford University Press, New York |year=1986 |isbn=0-19-853535-X |url=http://books.google.com/books?id=MbFBXyuxLKgC}}
 
== See also ==
*[[Loop space]]
*[[Loop algebra]]
*[[Quasigroup]]
 
[[Category:Topological groups]]
[[Category:Solitons]]

Latest revision as of 07:04, 28 March 2014

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