View source for Loop group

{{Short description|Loop space over a Lie group}}
{{for|groups of actors involved in re-recording movie dialogue during post-production (commonly known in the entertainment industry as "loop groups")|Dubbing (filmmaking)}}
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{{Group theory sidebar |Topological}}
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In [[mathematics]], a '''loop group''' (not to be confused with a [[Quasigroup#Loops|loop]]) is a [[group (mathematics)|group]] of [[loop (topology)|loop]]s in a [[topological group]] ''G'' with multiplication defined [[pointwise]].

==Definition==
In its most general form a loop group is a group of [[continuous function (topology)|continuous mappings]] from a [[manifold]] {{math|''M''}} to a topological group {{math|''G''}}.

More specifically,{{sfn|De Kerf|Bäuerle|Ten Kroode|1997}} let {{math|''M'' {{=}} ''S''<sup>1</sup>}}, the circle in the [[complex plane]], and let {{math|''LG''}} denote the [[Topological space|space]] of continuous maps {{math|''S''<sup>1</sup> → ''G''}}, i.e.
:<math>LG = \{\gamma:S^1 \to G|\gamma \in C(S^1, G)\},</math>

equipped with the [[compact-open topology]]. An element of {{math|''LG''}} is called a ''loop'' in {{math|''G''}}. 
Pointwise multiplication of such loops gives {{math|''LG''}} the structure of a topological group. Parametrize {{math|''S''<sup>1</sup>}} with {{mvar|θ}},
:<math>\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G,</math>

and define multiplication in {{math|''LG''}} by
:<math>(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).</math>

[[Associativity]] follows from associativity in {{math|''G''}}. The inverse is given by
:<math>\gamma^{-1}:\gamma^{-1}(\theta) \equiv \gamma(\theta)^{-1},</math>
and the identity by
:<math>e:\theta \mapsto e \in G.</math>

The space {{math|''LG''}} is called the '''free loop group''' on {{math|''G''}}. A loop group is any [[subgroup]] of the free loop group {{math|''LG''}}.

==Examples==

An important example of a loop group is the group 
:<math>\Omega G \,</math>

of based loops on {{math|''G''}}.  It is defined to be the [[kernel (algebra)|kernel]] of the evaluation map 
:<math>e_1: LG \to G,\gamma\mapsto \gamma(1)</math>,

and hence is a [[closed set|closed]] [[normal subgroup]] of {{math|''LG''}}. (Here, {{math|''e''<sub>1</sub>}} is the map that sends a loop to its value at <math>1 \in S^1</math>.) Note that we may embed {{math|''G''}} into {{math|''LG''}} 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''}} splits as a [[semi-direct product]], 
:<math>LG = \Omega G \rtimes G</math>.

We may also think of {{math|Ω''G''}} as the [[loop space]] on {{math|''G''}}.  From this point of view, {{math|Ω''G''}} is an [[H-space]] with respect to concatenation of loops. On the face of it, this seems to provide {{math|Ω''G''}} 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|Ω''G''}}, 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]].{{sfn|Terng|Uhlenbeck|2000}}

== See also ==
*[[Loop space]]
*[[Loop algebra]]
*[[Quasigroup]]

== Notes ==
{{reflist}}

== References ==
*{{citation |doi=10.1016/S0925-8582(97)80010-3 |chapter=Representations of loop algebras |title=Lie Algebras - Finite and Infinite Dimensional Lie Algebras and Applications in Physics |series=Studies in Mathematical Physics |date=1997 |volume=7 |pages=365–429 |isbn=978-0-444-82836-1 |editor1-first=E.A. |editor1-last=De Kerf |editor2-first=G.G.A. |editor2-last=Bäuerle |editor3-first=A.P.E. |editor3-last=Ten Kroode }}
*{{citation|mr=0900587|last1=Pressley|first1=Andrew|last2=Segal|first2=Graeme|authorlink2=Graeme Segal|title=Loop groups|series=Oxford Mathematical Monographs. Oxford Science Publications|publisher=[[Oxford University Press]]|location=New York|year=1986|isbn=978-0-19-853535-5|url=https://books.google.com/books?id=MbFBXyuxLKgC}}
*{{citation |last1=Terng |first1=Chuu-Lian |first2=Karen |last2=Uhlenbeck |title=Geometry of solitons |journal=Notices of the American Mathematical Society |volume=47 |issue=1 |date=2000 |pages=17–25 |url=https://www.ams.org/notices/200001/fea-terng.pdf }}

[[Category:Topological groups]]
[[Category:Solitons]]