# The Moment Maps in Diffeology

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*Patrick Iglesias-Zemmour*

This memoir presents a generalization of the moment maps to the category \(\{\)Diffeology\(\}\). This construction applies to every smooth action of any diffeological group \(\mathrm{G}\) preserving a closed 2-form \(\omega\), defined on some diffeological space \(\mathrm{X}\). In particular, that reveals a universal construction, associated to the action of the whole group of automorphisms \(\mathrm{Diff}(\mathrm{X},\omega)\). By considering directly the space of momenta of any diffeological group \(\mathrm{G}\), that is the space \(\mathscr{G}^*\) of left-invariant 1-forms on \(\mathrm{G}\), this construction avoids any reference to Lie algebra or any notion of vector fields, or does not involve any functional analysis. These constructions of the various moment maps are illustrated by many examples, some of them originals and others suggested by the mathematical literature.

#### Table of Contents

# Table of Contents

## The Moment Maps in Diffeology

- Introduction 18 free
- Chapter 1. Few words about diffeology 512 free
- Chapter 2. Diffeological groups and momenta 916
- Chapter 3. The paths moment map 1724
- Chapter 4. The 2-points moment map 2330
- Chapter 5. The moment maps 2532
- Chapter 6. The moment maps for exact 2-forms 2936
- Chapter 7. Functoriality of the moment maps 3138
- Chapter 8. The universal moment maps 3542
- Chapter 9. About symplectic manifolds 3946
- Chapter 10. The homogeneous case 4552
- Chapter 11. Examples of moment maps in diffeology 4754
- Bibliography 7178