The loop group 
of a Lie group 
is the group of smooth
maps
 |
from the unit circle to 
, with multiplication defined
pointwise. Other regularity classes, such as continuous and analytic loops, are used
when appropriate. A Sobolev loop is a map whose local coordinate representations
lie in a Sobolev space 
. Choosing a point on
the unit circle gives the based loop group
 |
where 
is the identity
element of 
.
When 
is a matrix Lie
group, a loop may be viewed as a matrix-valued function
of a spectral parameter. Loop groups and their factorizations
occur in the theory of harmonic maps, integrable
differential equations, and the DPW
method.
