Déformation et quantification par groupoïde des variétés toriques

Abstract : This thesis provides a notion of deformation quantization of Poisson manifolds, in a C*-algebraic framework, in relation with the use of groupoids. This theory is illustrated by examples, in particular the one of toric manifolds. The first part recalls some known facts on groupoids and their C*-algebras discovered in the past decades. The second part exposes the definitions of deformation and quantization used in the thesis, and their translation in terms of groupoids with the important notion of a deformation groupoid. A large class of subgroupoids of Lie groupoids is of this type. The main result of the thesis gives a condition on a manifold M endowed with an action of a torus Tn such that there is a deformation groupoid associated to it. It requires an action of Rn on a manifold containing the orbit space M/Tn ; the groupoid obtained is a subgroupoid of a groupoid of an action of a discrete group. We recover by this method known results for the quantization of Cn and the noncommutative torus and 4-spheres. The third part is an application of this result to the more complicated case of toric manifolds, using the unexpected discoveries of the 80's on the geometry of their moment map. This construction is the first exemple of quantization of toric manifolds, in a C*-algebraic framwork, even is the simplest cases (2-sphere and complex projective spaces). This provides in particular a C*-algebraic quantization of the 2-sphere and of complex projective spaces and answers the problem of the existence of such a quantization.
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Theses
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https://tel.archives-ouvertes.fr/tel-00001848
Contributor : Frédéric Cadet <>
Submitted on : Monday, October 21, 2002 - 4:53:48 PM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Long-term archiving on: Tuesday, September 11, 2012 - 6:10:28 PM

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  • HAL Id : tel-00001848, version 1

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Frédéric Cadet. Déformation et quantification par groupoïde des variétés toriques. Mathématiques [math]. Université d'Orléans, 2001. Français. ⟨tel-00001848⟩

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