# Loop group

Algebraic structure → Group theoryGroup theory |
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Group theory → Lie groupsLie groups |
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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 *M* to a topological group *G*.

More specifically,^{[1]} let *M* = *S*^{1}, the circle in the complex plane, and let *LG* denote the space of continuous maps *S*^{1} → *G*, i.e.

- [math]\displaystyle{ LG = \{\gamma:S^1 \to G|\gamma \in C(S^1, G)\}, }[/math]

equipped with the compact-open topology. An element of *LG* is called a *loop* in *G*.
Pointwise multiplication of such loops gives *LG* the structure of a topological group. Parametrize *S*^{1} with θ,

- [math]\displaystyle{ \gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G, }[/math]

and define multiplication in *LG* by

- [math]\displaystyle{ (\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta). }[/math]

Associativity follows from associativity in *G*. The inverse is given by

- [math]\displaystyle{ \gamma^{-1}:\gamma^{-1}(\theta) \equiv \gamma(\theta)^{-1}, }[/math]

and the identity by

- [math]\displaystyle{ e:\theta \mapsto e \in G. }[/math]

The space *LG* is called the **free loop group** on *G*. A loop group is any subgroup of the free loop group *LG*.

## Examples

An important example of a loop group is the group

- [math]\displaystyle{ \Omega G \, }[/math]

of based loops on *G*. It is defined to be the kernel of the evaluation map

- [math]\displaystyle{ e_1: LG \to G,\gamma\mapsto \gamma(1) }[/math],

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

- [math]\displaystyle{ 1\to \Omega G \to LG \to G\to 1 }[/math].

The space *LG* splits as a semi-direct product,

- [math]\displaystyle{ LG = \Omega G \rtimes G }[/math].

We may also think of Ω*G* as the loop space on *G*. From this point of view, Ω*G* is an H-space with respect to concatenation of loops. On the face of it, this seems to provide Ω*G* 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 Ω*G*, 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.^{[2]}

## Notes

- ↑ Bäuerle & de Kerf 1997
- ↑ Geometry of Solitons by Chuu-Lian Terng and Karen Uhlenbeck

## References

- Bäuerle, G.G.A; de Kerf, E.A. (1997).
*Finite and infinite dimensional Lie algebras and their application in physics*. Studies in mathematical physics.**7**. North-Holland. ISBN 978-0-444-82836-1. https://www.sciencedirect.com/bookseries/studies-in-mathematical-physics/vol/7/suppl/C. - Pressley, Andrew; Segal, Graeme (1986),
*Loop groups*, Oxford Mathematical Monographs. Oxford Science Publications, New York: Oxford University Press, ISBN 978-0-19-853535-5, https://books.google.com/books?id=MbFBXyuxLKgC

## See also

Original source: https://en.wikipedia.org/wiki/Loop group.
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