# 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$S^{1}\to G$ 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. The space $LG$ is called the free loop group on $G$ . A loop group is any subgroup of the free loop group $LG$ .

An important example of a loop group is the group

$\Omega G\,$ of based loops on $G$ . It is defined to be the kernel of the evaluation map

$e_{1}:LG\to G$ ,
$1\to \Omega G\to LG\to G\to 1$ .
$LG=\Omega G\rtimes G$ .

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