 Loop group

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 $2\pi$.

A **loop group** is a concept from mathematics, particularly in the fields of algebraic geometry, differential geometry, and mathematical physics. It typically refers to a specific kind of group associated with loops in a manifold, particularly in the context of Lie groups.