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Accueil du site > Projets de recherche financés en cours > Projet ERC Philosophie de la Gravitation Quantique Canonique > Evénements > Talk : Patrick Iglesias-Zemmour

Talk : Patrick Iglesias-Zemmour

Moment Maps in Diffeology

by Patrick Iglesias-Zemmour

November 27
room Kasimir Malevitch, 483A
University Paris Diderot

I will present a generalization of the Moment map to the category of diffeological spaces. This construction applies to any smooth action of a diffeological group on a diffeological space equipped with a closed 2-form (pre-symplectic structure). It applies very generally, without any presupposition as to the particular nature either of the space or of the group, and applies also to the group of all automorphisms of the pre-symplectic structure, revealing a universal Moment map associated with the two-form alone. This construction deliberately avoids generalizing Lie algebras, because it is a debatable concept in diffeology. Indeed we shall see that the only relevant space is that of the moments of the group, i.e. the space of left-invariant 1-forms, which itself is conceptually at home in diffeology. With this construction, all essential objects related to pre-symplectic actions, generalized to the group of automorphisms, can be recovered, e.g. the obstruction to the Hamiltonian nature of the action, the Souriau cohomology class, etc. Remarkably, this does not appeal to any differential equation. This shows how simplifying and conceptually significant this generalization is. I will try to give an example or two of Moment calculation in an infinite dimensional case and in a singular case.

Room Malevitch, 483A, building Condorcet,
4, rue Elsa Morante, 75013 Paris.
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