Generators for Rational Loop Groups and Geometric Applications

2008/03/01 by Neil Donaldson, Daniel J. F. Fox, Donaldson, Neil +4
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #math.DG

paper · pdf · doi:10.48550/arxiv.0803.0029

arxiv created 2008/03/01 · openalex publication_date 2008/03/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down 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. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G₂/SO(4) to which our theory applies.

Citations

• Conformally flat submanifolds in spheres and integrable systems
• Transformations of flat Lagrangian immersions and Egoroff nets
• Calibrated geometries

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