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The moment map in diffeology

Patrick Iglesias-Zemmour


The moment map has been introduced in the middle of the 20th century by J.-M. Souriau as a main tool for dealing with symmetries in symplectic geometry. It involves usually a symplectic (or pre-symplectic) manifold, a Lie group (the group of symmetries) and an action of the Lie group on the manifold preserving the (pre) symplectic form. The moment map itself (when it exists) is a map from the manifold to the dual of the Lie algebra of the Lie group. This map is related to the generalization of Noether's theorem, classical model of elementary particles and their quantization (in the framework of geometric quantization), classification of manifolds with symmetries and more... Since more a decade now, in a lot of constructions involving infinite dimensional spaces and groups, mathematicians play with objects which look like symplectic forms and moment maps. In general these constructions are heuristic and each example has its own interpretation of what is symplectic and what is the moment map. Sometimes, mathematicians use heavy functional analysis tools in order to give to these constructions a more serious content. These questions related to generalization of moment map do not arise just for infinite dimensional space but also for what is often regarded as singular spaces, like orbifolds or more serious singular quotients. In this talk, I will describe a general and formal construction of the moment maps (and related objects) for the category of diffeological spaces equipped with a closed 2-form. Since the category {Diffeology} includes in one movement the various cases of infinite dimensional spaces, manifolds, and singular spaces, this construction is a common formal answer for all of these situations. In spite of the light (but not trivial) nature of the category {Diffeology} we shall see in particular that this construction is powerful enough to retrieve (and give a formal meaning to) the usual examples of the mathematical folklore. We shall see also how the diffeological approach, because it needs to be rewritten in a complete co-variant language, extends and simplifies enormously the theory of moment maps, even the classical construction, making crystal-clear all the obstructions, cohomologies and other subtleties appearing in the theory.

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