Loop group

In mathematics, a loop group is a group of loops in a topological group "G" with multiplication defined pointwise. Specifically, let $$LG denote the space of continuous maps $$S^1 o G 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. The space $$LG is called the free loop group on $$G. A loop group is any subgroup of the free loop group $$LG.

An important example of a loop group is the group $$Omega G of based loops on $$G. It is defined to be the kernel of the evaluation map $$e_1: LG o G, and hence is a closed normal subgroup of $$LG. (Here, $$e_1 is the map that sends a loop to its value at $$1.) Note that we may embed $$G into $$LG as the subgroup of constant loops. Consequently, we arrive at a split exact sequence $$1 o Omega G o LG o G o 1. The space $$LG splits as a semi-direct product, $$LG = Omega G ltimes G.

We may also think of $$Omega G as the loop space on $$G. From this point of view, $$Omega G is an "H"-space with respect to concatenation of loops. On the face of it, this seems to provide $$Omega G with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of $$Omega G, these maps are interchangeable.

ee also

*loop space
*loop algebra