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The Moment Maps in Diffeology
Patrick Iglesias-Zemmour, CNRS, Marseille, France, and Hebrew University of Jerusalem, Israel
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Memoirs of the American Mathematical Society
2010; 72 pp; softcover
Volume: 207
ISBN-10: 0-8218-4709-0
ISBN-13: 978-0-8218-4709-1
List Price: US$64
Individual Members: US$38.40
Institutional Members: US$51.20
Order Code: MEMO/207/972
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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 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

  • Introduction
  • Few words about diffeology
  • Diffeological groups and momenta
  • The paths moment map
  • The 2-points moment map
  • The moment maps
  • The moment maps for exact 2-forms
  • Functoriality of the moment maps
  • The universal moment maps
  • About symplectic manifolds
  • The homogeneous case
  • Examples of moment maps in diffeology
  • Bibliography
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