Loop group

In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group G with multiplication defined pointwise.

Definition

In its most general form a loop group is a group of continuous mappings from a manifold Template:Math to a topological group Template:Math.

More specifically,Template:Sfn let Template:Math, the circle in the complex plane, and let Template:Math denote the space of continuous maps Template:Math, 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 Template:Math is called a loop in Template:Math. Pointwise multiplication of such loops gives Template:Math the structure of a topological group. Parametrize Template:Math with Template:Mvar,

<math>\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G,</math>

and define multiplication in Template:Math by

<math>(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta).</math>

Associativity follows from associativity in Template:Math. 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 Template:Math is called the free loop group on Template:Math. A loop group is any subgroup of the free loop group Template:Math.

Examples

An important example of a loop group is the group

<math>\Omega G \,</math>

of based loops on Template:Math. It is defined to be the kernel of the evaluation map

<math>e_1: LG \to G,\gamma\mapsto \gamma(1)</math>,

and hence is a closed normal subgroup of Template:Math. (Here, Template:Math is the map that sends a loop to its value at <math>1 \in S^1</math>.) Note that we may embed Template:Math into Template:Math 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 Template:Math splits as a semi-direct product,

<math>LG = \Omega G \rtimes G</math>.

We may also think of Template:Math as the loop space on Template:Math. From this point of view, Template:Math is an H-space with respect to concatenation of loops. On the face of it, this seems to provide Template:Math 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 Template:Math, 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.Template:Sfn

See also

• Loop space
• Loop algebra
• Quasigroup

References

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