Electronic ISBN:  9781470405861 
Product Code:  MEMO/207/972.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 207; 2010; 72 ppMSC: Primary 53;
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 2form \(\omega\), 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 leftinvariant 1forms 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

Chapters

Introduction

1. Few words about diffeology

2. Diffeological groups and momenta

3. The paths moment map

4. The 2points moment map

5. The moment maps

6. The moment maps for exact 2forms

7. Functoriality of the moment maps

8. The universal moment maps

9. About symplectic manifolds

10. The homogeneous case

11. Examples of moment maps in diffeology


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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 2form \(\omega\), 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 leftinvariant 1forms 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.

Chapters

Introduction

1. Few words about diffeology

2. Diffeological groups and momenta

3. The paths moment map

4. The 2points moment map

5. The moment maps

6. The moment maps for exact 2forms

7. Functoriality of the moment maps

8. The universal moment maps

9. About symplectic manifolds

10. The homogeneous case

11. Examples of moment maps in diffeology