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.
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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 , every loop has the form , where is its winding number and is a smooth real periodic function determined up to addition of an integer multiple of .