Rankin–Cohen brackets and formal quantization

Abstract

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [5]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [11] 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 [9], we reconstruct a universal deformation formula of the Hopf algebra H1 associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket RC1 defines a noncommutative Poisson structure for an arbitrary H1 action. *

Rankin–Cohen Brackets and Associativity

“…The same idea, to wit the fact that the Rankin-Cohen brackets do define an associative deformation of the graded ring of modular forms were discussed in [4,6,3,16,17,22].…”

Universal deformation formulae, symplectic Lie groups and symmetric spaces

“…Our construction differs radically from the (formal) universal deformation formula of Giaquinto and Zhang [1998]: It was shown in [Bieliavsky et al 2005b] that the (maximal) invariance diffeomorphism group of GiaquintoZhang star product on ax + b is isomorphic to Sp(1, ‫)ޒ‬ × ‫ޒ‬ 2 , while in our case the corresponding maximal invariance group is the automorphism group of the underlying symplectic symmetric space -in the two-dimensional case, the solvable group SO(1, 1) × ‫ޒ‬ 2 .…”

Symplectic Connections

“…For instance, the solution for the general solvable symmetric case has led to several universal deformation formulae for actions of various classes of solvable Lie groups [10,9,6]. Applications in noncommutative geometry (non-commutative causal black holes [7]) as well as in analytic number theory (Rankin-Cohen brackets on modular forms [11]) have been developed.…”