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Deformation quantization of pseudo-symplectic (Poisson) groupoids. (English) Zbl 1119.53061

In the paper under review the author introduces a new kind of groupoid – a pseudo-étale groupoid – which provides new examples of noncommutative Poisson algebras. The deformation quantization of these algebras is investigated. It is shown that the noncommutative Poisson algebra introduced by the pseudo-étale groupoid can be formally deformation quantized.
The starting point of the presentation is the notion of Poisson structure on an associative algebra [J. Block and E. Getzler, Quantization of foliations, in S. Catto et al., Differential geometric methods in theoretical physics. Proceedings of the 20th international conference, June 3–7, 1991, New York City, NY, USA. Vol. 1–2. Singapore: World Scientific. 471–487 (1992; Zbl 0812.58028); P. Xu, Am. J. Math. 116, 101–125 (1994; Zbl 0797.58012)]. The smooth groupoid algebra of a Lie groupoid, which consists of smooth functions with convolution product, is an important example of a noncommutative differentiable manifold. The groupoid algebra is defined by a Haar system [A. L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics 170, Boston, MA: Birkhäuser (1999; Zbl 0913.22001)] and use is made of Crainic’s and Moerdijk’s definition of a smooth groupoid algebra [M. Crainic and I. Moerdijk, J. Reine Angew. Math. 521, 25–46 (2000; Zbl 0954.22002)].
It is shown that a pseudo-regular Poisson groupoid with a given invariant connection naturally defines a noncommutative Poisson structure. Following M. Kontsevich [Lett. Math. Phys. 66, 157–216 (2003; Zbl 1058.53065)], an “equivariant formality theorem” with groupoid action is discussed. Next the formal deformation quantization of a noncommutative Poisson algebra à la Fedosov is defined [B. Fedosov, Deformation quantization and index theory, Mathematical Topics 9, Berlin: Akademie Verlag (1996; Zbl 0867.58061)]. A \(\star\) -product is introduced on the groupoid algebra and its associativity is proved. In order to make similar constructions for deformation quantization in the case of the pseudo-Poisson groupoid with that already made for the pseudo-symplectic groupoid, a “quasi-connection” is introduced. The construction of A. S. Cattaneo, G. Felder and L. Tomassini [Duke Math. J. 115, 329–352 (2002; Zbl 1037.53063)] is used in the present situation, applying Kontsevich’s local formality theorem. The main result is that for a pseudo-Poisson groupoid, there is always a formal deformation quantization of the noncommutative Poisson algebra. At the end of the paper the existence of closed deformation quantization of a pseudo-symplectic (Poisson) groupoid with an invariant measure is defined and proved. Also Rieffel’s strict deformation quantization [M. A. Rieffel, Deformation quantization for actions of \(\mathbb{R}^d\), Mem. Am. Math. Soc. 506 (1993; Zbl 0798.46053)] of noncommutative Poisson algebras is discussed.

MSC:

53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
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