Rankin-Cohen brackets and formal quantization

2005/06/24 by Pierre Bieliavsky, Xiang Tang, Bieliavsky, Pierre +3
Mathematics · #46L87 #58H05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.math/0506506

openalex publication_date 2005/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \citecm:deformation. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \citez:deformation 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 \citet1:def-gpd, we reconstruct a universal deformation formula of the Hopf algebra \calh₁ 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 \calh₁ action.

Citations

• Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

Related

• Poisson bracket, deformed bracket and gauge group actions in Kontsevich deformation quantization
• Deformation Quantisation via Kontsevich Formality Theorem
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• Deformation Quantization: Genesis, Developments and Metamorphoses