Editing Loop group (section)

==Definition==
In its most general form a loop group is a group of [[continuous function (topology)|continuous mappings]] from a [[manifold]] {{math|''M''}} to a topological group {{math|''G''}}.

More specifically,{{sfn|De Kerf|Bäuerle|Ten Kroode|1997}} let {{math|''M'' {{=}} ''S''<sup>1</sup>}}, the circle in the [[complex plane]], and let {{math|''LG''}} denote the [[Topological space|space]] of continuous maps {{math|''S''<sup>1</sup> → ''G''}}, i.e.
:<math>LG = \{\gamma:S^1 \to G|\gamma \in C(S^1, G)\},</math>

equipped with the [[compact-open topology]]. An element of {{math|''LG''}} is called a ''loop'' in {{math|''G''}}. 
Pointwise multiplication of such loops gives {{math|''LG''}} the structure of a topological group. Parametrize {{math|''S''<sup>1</sup>}} with {{mvar|θ}},
:<math>\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G,</math>

and define multiplication in {{math|''LG''}} by
:<math>(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).</math>

[[Associativity]] follows from associativity in {{math|''G''}}. The inverse is given by
:<math>\gamma^{-1}:\gamma^{-1}(\theta) \equiv \gamma(\theta)^{-1},</math>
and the identity by
:<math>e:\theta \mapsto e \in G.</math>

The space {{math|''LG''}} is called the '''free loop group''' on {{math|''G''}}. A loop group is any [[subgroup]] of the free loop group {{math|''LG''}}.