Loop group

In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise.

In its most general form a loop group is a group of 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 $S 1 \to G$ , i.e.

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 $θ$ ,

and define multiplication in $LG$ by

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

and the identity by

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

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

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

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

We may also think of $Ω G$ as the loop space on $G$ . From this point of view, $Ω G$ is a 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.

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