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Rankin-Cohen brackets and formal quantization. (English) Zbl 1123.53049
Summary: We use the theory of deformation quantization to understand the results of A. Connes and H. Moscovici [Mosc. Math. J. 4, No. 1, 111–130 (2004; Zbl 1122.11024)]. We use Fedosov’s method of deformation quantization of symplectic manifolds to reconstruct Zagier’s deformation [D. Zagier, Proc. Indian Acad. Sci., Math. Sci. 104, No. 1, 57–75 (1994; Zbl 0806.11022)] of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author [X. Tang, Geom. Funct. Anal. 16, No. 3, 731–766 (2006; Zbl 1119.53061)], we reconstruct a universal deformation formula of the Hopf algebra \(\mathcal {H}_1\)associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket \(RC_{1}\) defines a noncommutative Poisson structure for an arbitrary \(\mathcal {H}_1\) action.

MSC:
53D55 Deformation quantization, star products
46L87 Noncommutative differential geometry
11F32 Modular correspondences, etc.
16T05 Hopf algebras and their applications
46L65 Quantizations, deformations for selfadjoint operator algebras
58H05 Pseudogroups and differentiable groupoids
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References:
[1] P. Bieliavsky, X. Tang, Y. Yao, in preparation
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