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Loop group

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{{Group theory sidebar |Topological}}

{{Lie groups |Other}}

In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group G with multiplication defined pointwise.

Definition

In its most general form a loop group is a group of 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 {{=}} S1}}, the circle in the complex plane, and let {{math|LG}} denote the space of continuous maps {{math|S1 → G}}, i.e.

:LG = \{\gamma:S^1 \to G|\gamma \in C(S^1, G)\},

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|S1}} with {{mvar|θ}},

:\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G,

and define multiplication in {{math|LG}} by

:(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).

Associativity follows from associativity in {{math|G}}. The inverse is given by

:\gamma^{-1}:\gamma^{-1}(\theta) \equiv \gamma(\theta)^{-1},

and the identity by

:e:\theta \mapsto e \in G.

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

:\Omega G \,

of based loops on {{math|G}}. It is defined to be the kernel of the evaluation map

:e_1: LG \to G,\gamma\mapsto \gamma(1),

and hence is a closed normal subgroup of {{math|LG}}. (Here, {{math|e1}} is the map that sends a loop to its value at 1 \in S^1.) Note that we may embed {{math|G}} into {{math|LG}} as the subgroup of constant loops. Consequently, we arrive at a split exact sequence

:1\to \Omega G \to LG \to G\to 1.

The space {{math|LG}} splits as a semi-direct product,

:LG = \Omega G \rtimes G.

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

See also

Notes

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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