Rankin-Cohen brackets and formal quantization

A Survey on Rankin-Cohen Deformations

This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms.

Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry

We settle in this paper a question left open in our paper ``Modular Hecke algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to associative deformations.

Rankin-Cohen Deformations and Representation Theory

In this paper, we use the unitary representation theory of $SL_2(\mathbb R)$ to understand the Rankin-Cohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.

Differential symmetry breaking operators. I-Genreal theory and F-method. II-Rankin-Cohen Operators for Symmetric Pairs

Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry b…

代数曲线典范锥的量子化

Quantization of canonical cones of algebraic curves

本文利用形式伪微分算子对代数曲线典范锥上的函数分次代数进行量子化，并在 Poincaré 单值化情形下借助 Rankin-Cohen 括号给出等价构造，还描述了有理曲线时的量子代数表示。

Associative algebra deformations of the Connes-Moscovici's Hopf algebra $\mathcal{H}_1$

We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence.