# Loop group

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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${\displaystyle S^{1}\to G}$ equipped with the compact-open topology. An element of ${\displaystyle LG}$ is called a loop in G. Pointwise multiplication of such loops gives ${\displaystyle LG}$ the structure of a topological group. The space ${\displaystyle LG}$ is called the free loop group on ${\displaystyle G}$. A loop group is any subgroup of the free loop group ${\displaystyle LG}$.

An important example of a loop group is the group

${\displaystyle \Omega G\,}$

of based loops on ${\displaystyle G}$. It is defined to be the kernel of the evaluation map

${\displaystyle e_{1}:LG\to G}$,

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

${\displaystyle 1\to \Omega G\to LG\to G\to 1}$.

The space ${\displaystyle LG}$ splits as a semi-direct product,

${\displaystyle LG=\Omega G\rtimes G}$.

We may also think of ${\displaystyle \Omega G}$ as the loop space on ${\displaystyle G}$. From this point of view, ${\displaystyle \Omega G}$ is an H-space with respect to concatenation of loops. On the face of it, this seems to provide ${\displaystyle \Omega G}$ 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 ${\displaystyle \Omega G}$, 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.[1]

## Notes

1. Geometry of Solitons by Chuu-Lian Terng and Karen Uhlenbeck

## References

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