Generating Rational Loop Groups with Noncompact Reality Conditions

Abstract

We find generators for the full rational loop group of ${\rm GL}(n,\mathsf{C})$ as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form ${\rm GL}(n,\mathsf{R})$. We calculate the dressing action of some of those generators on the positive loop group, and apply this to the ZS-AKNS flows and the $n$-dimensional system associated to ${\rm GL}(n,\mathsf{R})/{\rm O}(n)$.