System of imprimitivity

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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

$\mathrm{\Omega }G$

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

$e_1:LG\to G$,

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

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

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

$LG=\mathrm{\Omega }G⋊G$.

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

Notes

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References

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See also

• Loop space
• Loop algebra
• Quasigroup