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, let Template:Math, the circle in the complex plane, and let Template:Math denote the space of continuous maps Template:Math, i.e.

$LG=\{\gamma :S^1\to G|\gamma \in C(S^1,G)\},$

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,

$\gamma :\theta \in S^1\mapsto \gamma (\theta )\in G,$

and define multiplication in Template:Math by

$({\gamma }_1{\gamma }_2)(\theta )\equiv {\gamma }_1(\theta ){\gamma }_2(\theta ).$

Associativity follows from associativity in Template:Math. The inverse is given by

${\gamma }^{-1}:{\gamma }^{-1}(\theta )\equiv \gamma (\theta {)}^{-1},$

and the identity by

$e:\theta \mapsto e\in G.$

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

$\mathrm{\Omega }G$

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

$e_1:LG\to G,\gamma \mapsto \gamma (1)$,

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

$LG=\mathrm{\Omega }G⋊G$.

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.

References

• Template:Cite book
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See also

• Loop space
• Loop algebra
• Quasigroup