In mathematics, a loop group 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 to a topological group .
More specifically, let , the circle in the complex plane, and let denote the space of continuous maps , i.e.
equipped with the compact-open topology. An element of is called a loop in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with ,
and define multiplication in by
Associativity follows from associativity in . The inverse is given by
and the identity by
The space is called the free loop group on . A loop group is any subgroup of the free loop group .
Examples
An important example of a loop group is the group
of based loops on . It is defined to be the kernel of the evaluation map
,
and hence is a closed normal subgroup of . (Here, is the map that sends a loop to its value at .) Note that we may embed into as the subgroup of constant loops. Consequently, we arrive at a split exact sequence .
The space splits as a semi-direct product, .
We may also think of as the loop space on . From this point of view, is an H-space with respect to concatenation of loops. On the face of it, this seems to provide 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 , 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.