Generating rational loop groups with noncompact reality conditions

TL;DR

This paper identifies generators for the rational loop group of GL(n,C) and its noncompact real form GL(n,R), analyzing their dressing actions and applications to integrable systems.

Contribution

It introduces generators for the rational loop group of GL(n,C) and its noncompact real form, enabling new insights into their structure and actions.

Findings

Generated the full rational loop group of GL(n,C)

Derived generators for the noncompact real form GL(n,R)

Applied dressing actions to integrable systems like ZS-AKNS flows

Abstract

We find generators for the full rational loop group of GL(n,C) as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form GL(n,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 GL(n,R)/O(n).