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Classification of multiplicity free quasi-Hamiltonian manifolds

Friedrich Knop

TL;DR

The paper extends Delzant-type classifications to multiplicity free quasi-Hamiltonian manifolds by developing a local-to-global framework based on affine and local root systems, a band/gerbe structure, and local models from affine spherical varieties. The core result shows that compact MF twisted quasi-Hamiltonian manifolds for simply connected compact groups are classified by convex spherical pairs $(bP,oldsymbol extLambda)$, with existence and uniqueness guaranteed via a vanishing cohomology theorem and gluing of local data. The authors generalize earlier Hamiltonian classifications to the quasi-Hamiltonian setting, including twists, and provide extensive examples such as doubles, spinning spheres, and quaternionic Grassmannians, as well as surjective momentum-map cases. This work yields a unified, constructive framework for MF quasi-Hamiltonian geometry and offers a toolkit for identifying new MF manifolds from spherical pair data.

Abstract

A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.

Classification of multiplicity free quasi-Hamiltonian manifolds

TL;DR

The paper extends Delzant-type classifications to multiplicity free quasi-Hamiltonian manifolds by developing a local-to-global framework based on affine and local root systems, a band/gerbe structure, and local models from affine spherical varieties. The core result shows that compact MF twisted quasi-Hamiltonian manifolds for simply connected compact groups are classified by convex spherical pairs , with existence and uniqueness guaranteed via a vanishing cohomology theorem and gluing of local data. The authors generalize earlier Hamiltonian classifications to the quasi-Hamiltonian setting, including twists, and provide extensive examples such as doubles, spinning spheres, and quaternionic Grassmannians, as well as surjective momentum-map cases. This work yields a unified, constructive framework for MF quasi-Hamiltonian geometry and offers a toolkit for identifying new MF manifolds from spherical pair data.

Abstract

A quasi-Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify compact, multiplicity free, twisted quasi-Hamiltonian manifolds for simply connected, compact Lie groups. Thereby, we recover old and find new examples of these structures.
Paper Structure (13 sections, 37 theorems, 122 equations)

This paper contains 13 sections, 37 theorems, 122 equations.

Table of Contents

  1. Local root systems and cohomology
  2. Affine root systems
  3. Local root systems
  4. The automorphism sheaf
  5. The vanishing theorem
  6. Classification of multiplicity free quasi-Hamiltonian manifolds
  7. Quasi-Hamiltonian manifolds
  8. Twisted conjugacy classes
  9. The local structure of quasi-Hamiltonian manifolds
  10. Multiplicity free manifolds
  11. Local models
  12. Classification of multiplicity free quasi-Hamiltonian manifolds
  13. Examples

Key Result

Proposition 1

Let $(\Phi(*),\Lambda)$ be a local root system on $\mathcal{P}\subseteq\mathfrak{a}$ and let $W\subseteq M(\mathfrak{a})$ be the subgroup generated by all local Weyl groups $W(x)$ of $\Phi(x)$ with $x\in\mathcal{P}$. Assume: Then the local root system is trivial.

Theorems & Definitions (94)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Remark 1
  • Definition 5
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 84 more