Loop group

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 $S^1\; o\; G$ 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 $e\_1:\; LG\; o\; G$, 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 $1\; o\; Omega\; G\; o\; LG\; o\; G\; o\; 1$. The space $LG$ splits as a semi-direct product, $LG\; =\; Omega\; G\; ltimes\; G$.

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.

ee also

*loop space
*loop algebra