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Quantization of the cotangent bundle via the tangent groupoid. (English) Zbl 0987.53035

Grabowski, Janusz (ed.) et al., Poisson geometry. Stanisław Zakrzewski in memoriam. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 51, 43-53 (2000).
From the introduction: In the present paper we exploit some ideas by A. Connes [“Noncommutative Geometry”, Academic Press, London (1994; Zbl 0818.46076)] and N. P. Landsman [J. Geom. Phys. 12, 93-132 (1993; Zbl 0789.58081)] to construct a quantization procedure for a cotangent bundle. Actually, we report and generalize the construction presented in a recent work [the authors, E. Follana, J. M. García-Bondía, A. Rivero and J. C. Varilly, J. Geom. Phys. 32, 79-96 (1999; Zbl 0961.53047)] to a bigger set of quantizations. We will see that this construction allows us to define a strict quantization in a very large set of cases. We will also study some physical properties of the corresponding quantizations, and we will see that in some particular cases, the construction can be used to define a strict deformation quantization.
The paper is divided into two parts: a brief exposition of the geometrical structures we are handling (the tangent and normal groupoids) which covers the first section, and the use of these groupoids to define a strict quantization, in sections two (the general case) and three (a short comment about the construction of a strict deformation quantization in exponential manifolds).
For the entire collection see [Zbl 0936.00035].

MSC:

53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
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