Loop group
= Loop group
{title2=$LG=C^\infty(S^1,G)$}
{wiki}
The loop group of a <Lie group> is the group of smooth maps from the circle to that group, with pointwise multiplication and the smooth-function topology. Rotation of the source circle acts by group automorphisms. For $G=U(1)$, every loop has the form $z^w e^{ih(z)}$, where $w\in\mathbb Z$ is its <winding number> and $h$ is a smooth real periodic function determined up to addition of an integer multiple of $2\pi$.