Loop group

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, let $M = S^{1}$, the circle in the complex plane, and let $LG$ denote the space of continuous maps $S^{1} → G$, i.e.
 * $$LG = \{\gamma:S^1 \to G|\gamma \in C(S^1, G)\},$$

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. Parametrize $S^{1}$ with $θ$,
 * $$\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G,$$

and define multiplication in $LG$ by
 * $$(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).$$

Associativity follows from associativity in $G$. 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 $LG$ 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
 * $$\Omega G \,$$

of based loops on $G$. 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 $LG$. (Here, $e_{1}$ is the map that sends a loop to its value at $$1 \in S^1$$.) Note that we may embed $G$ into $LG$ 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 $LG$ splits as a semi-direct product,
 * $$LG = \Omega G \rtimes G$$.

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