Loop group

Parametrize S1 with θ, and define multiplication in LG by Associativity follows from associativity in G. The inverse is given by and the identity by The space LG is called the free loop group on G. A loop group is any subgroup of the free loop group LG.

Note that we may embed G into LG as the subgroup of constant loops.

Consequently, we arrive at a split exact sequence The space LG splits as a semi-direct product, We may also think of ΩG as the loop space on G. From this point of view, ΩG is an H-space with respect to concatenation of loops.

Thus, in terms of the homotopy theory of ΩG, 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.