Rankin–Cohen brackets and formal quantization

Abstract

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [A. Connes, H. Moscovici, Rankin–Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (1) (2004) 111–130, 311]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [D. Zagier, Modular forms and differential operators, in: K.G. Ramanathan Memorial Issue, Proc. Indian Acad. Sci. Math. Sci. 104 (1) (1994) 57–75] 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, Deformation quantization of pseudo (symplectic) Poisson groupoids, Geom. Funct. Anal. 16 (3) (2006) 731–766], we reconstruct a universal deformation formula of the Hopf algebra 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 H 1 action.