nLab Rankin-Cohen bracket

Overview and definition

Let $\Gamma$ be a congruence subgroup of $SL(2,\mathbf{Z})$ and $\mathcal{M}(\Gamma)$ the graded (by the weight) algebra of modular forms with respect to $\Gamma$. All bidifferential operators which leave that space invariant are linear combinations of Rankin-Cohen brackets $[-,-]_n\colon(f, g)\mapsto [f,g]_n$. By definition, $n$-th bracket between elements $f\in\mathcal{M}(\Gamma)_{2k}$ and $g\in\mathcal{M}(\Gamma)_{2l}$ is given by the formula

$[f,g]_n := \sum_{r=0}^n (-1)^r\binom{n+2k-1}{n-r}\binom{n+2l-1}{r}f^{(r)}g^{(n-r)}\,\in \,\mathcal{M}(\Gamma)_{2k+2l+2n}$

where $f^{(r)} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial z}\right)^r f$. They are directly related to invariant differential operators used to produce new $sl(2)$-invariant bilinear forms from old ones, so called transvectants found by Gordan,

• P. Gordan, Das Zerfallen der Curven in gerade Linien, Math. Ann., 45 (1894), pp. 411-427 doi

Moscovici and Connes have constructed a sequence of Hopf algebras $\mathcal{H}_q$ related to geometry of foliations. Hopf algebra $\mathcal{H}_q$ has deformations which may be given by universal deformation formulas, or in other words, by Drinfeld twists which are power series in formal variable with unit free term. These 2-cocycles have the structure appearing in Rankin-Cohen brackets and are called Rankin-Cohen deformations and are akin in structure to what is in quantum group context known as Jordanian twist?s, coming from the work of Gurevich on (generalized) Jordanian R-matrices, and of Ogievetsky, Coll-Gerstenhaber-Giaquinto and in a closer symmetrized form by Giaquinto and Zhang. An isomorphism between (reduced) Rankin-Cohen deformation and Jordanian deformation has been exhibited by Samsonov.

Literature

• R. A. Rankin, The construction of automorphic forms from the derivatives of a given form, J. Indian Math.Soc., (N.S.) 20 (1956) 103–116
• H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math.Ann. 217 (1975) 271-285
• P. Cohen, Yu. Manin, D. Zagier, Automorphic pseudodifferential operators, Algebraic aspects of integrable systems, 17–47, Progr. Nonlinear Differential Equations Appl.26, Birkauser 1997
• Alain Connes, Henri Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4:1 (2004) 111–130
• R. Rochberg, X. Tang, Y. Yao, A survey of Rankin-Cohen deformations, Perspectives on Noncommutative Geometry, Fields Inst. Commun. 61:7 (2011) arXiv:0909.4364
• P. Bieliavsky, X. Tang, Y. Yao, Rankin–Cohen brackets and formal quantization, Adv. Math. 212:1 (2007) 293-314 doi
• M. Samsonov, Quantization of semi-classical twists and noncommutative geometry, Lett. Math. Phys. 75 (2006) 63–77 doi
• Y.–J. Choie, W. Eholzer, Rankin–Cohen operators for Jacobi and Siegel forms, J. Number Theory 68 (1998) 160–177