Generators for rational loop groups

Abstract:

Uhlenbeck proved that a set of simple elements generates the group of rational loops in $\mathrm {GL}(n,\mathbb {C})$ that satisfy the $\mathrm {U}(n)$-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neo-classical groups and most of their symmetric spaces. We also obtain explicit dressing and permutability formulae.