# Projective loops generate rational loop groups

## TL;DR

This paper proves that projective loops alone are sufficient to generate rational loop groups for certain classical groups, simplifying previous assumptions that nilpotent loops were necessary.

## Contribution

It demonstrates that projective loops alone generate rational loop groups of GL(n,C), GL(n,R), and U(p,q), challenging prior beliefs about the need for nilpotent loops.

## Key findings

- Projective loops generate rational loop groups for GL(n,C), GL(n,R), and U(p,q).
- Nilpotent loops are not necessary for generating these groups.
- Simplifies the understanding of the structure of rational loop groups.

## Abstract

Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational loop groups with noncompact reality conditions. We prove a somewhat unexpected theorem that projective loops are enough to generate the rational loop groups of GL(n,C), GL(n,R), and U(p, q).