Loop group

In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. In its most general form a loop group is a group of mappings from a manifold M to a topological group G. More specifically, let M = S1, the circle in the complex plane, and let LG denote the space of continuous maps S1 → G, i.e. equipped with the compact-open topology. An element of LG is called a loop in G. Pointwise multiplication of such loops gives LG the structure of a topological group. Parametrize S1 with θ, and define multiplication in LG by Associativity follows from associativity in G. The inverse is given by