Loop group: Difference between revisions
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{{for|groups of actors involved in re-recording movie dialogue during post-production (commonly known in the entertainment industry as "loop groups")|Dubbing (filmmaking)}} |
{{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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In [[mathematics]], a '''loop |
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>. |
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An important example of a loop |
An important example of a loop group is the group |
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:<math>\Omega G \,</math> |
:<math>\Omega G \,</math> |
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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. |
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. |
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Loop |
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> |
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== Notes == |
== Notes == |
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== References == |
== References == |
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*{{citation |mr=0900587 |last=Pressley |first=Andrew |last2=Segal |first2=Graeme |title=Loop |
*{{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}} |
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== See also == |
== See also == |
Revision as of 05:14, 28 December 2013
Algebraic structure → Group theory Group theory |
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Lie groups and Lie algebras |
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In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, let LG denote the space of continuous maps equipped with the compact-open topology. An element of is called a loop in G. Pointwise multiplication of such loops gives the structure of a topological group. The space is called the free loop group on . A loop group is any subgroup of the free loop group .
An important example of a loop group is the group
of based loops on . It is defined to be the kernel of the evaluation map
- ,
and hence is a closed normal subgroup of . (Here, is the map that sends a loop to its value at .) Note that we may embed into as the subgroup of constant loops. Consequently, we arrive at a split exact sequence
- .
The space splits as a semi-direct product,
- .
We may also think of as the loop space on . From this point of view, is an H-space with respect to concatenation of loops. On the face of it, this seems to provide with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of , these maps are interchangeable.
Loop groups were used to explain the phenomenon of Bäcklund transforms in soliton equations by Chuu-Lian Terng and Karen Uhlenbeck.[1]
Notes
- ^ Geometry of Solitons by Chuu-Lian Terng and Karen Uhlenbeck
References
- Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, ISBN 0-19-853535-X, MR 0900587