Editing Loop group (section)

==Examples==

An important example of a loop group is the group 
:<math>\Omega G \,</math>

of based loops on {{math|''G''}}.  It is defined to be the [[kernel (algebra)|kernel]] of the evaluation map 
:<math>e_1: LG \to G,\gamma\mapsto \gamma(1)</math>,

and hence is a [[closed set|closed]] [[normal subgroup]] of {{math|''LG''}}. (Here, {{math|''e''<sub>1</sub>}} is the map that sends a loop to its value at <math>1 \in S^1</math>.) Note that we may embed {{math|''G''}} into {{math|''LG''}} as the subgroup of constant loops.  Consequently, we arrive at a [[split exact sequence]] 
:<math>1\to \Omega G \to LG \to G\to 1</math>.

The space {{math|''LG''}} splits as a [[semi-direct product]], 
:<math>LG = \Omega G \rtimes G</math>.

We may also think of {{math|Ω''G''}} as the [[loop space]] on {{math|''G''}}.  From this point of view, {{math|Ω''G''}} is an [[H-space]] with respect to concatenation of loops. On the face of it, this seems to provide {{math|Ω''G''}} with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are [[homotopy|homotopic]]. Thus, in terms of the homotopy theory of {{math|Ω''G''}}, these maps are interchangeable.

Loop groups were used to explain the phenomenon of [[Bäcklund transform]]s in [[soliton]] equations by [[Chuu-Lian Terng]] and [[Karen Uhlenbeck]].{{sfn|Terng|Uhlenbeck|2000}}