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Rankin-Cohen bracket

Overview and definition

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

[f,g] n:= r=0 n(1) r(n+2k1nr)(n+2l1r)f (r)g (nr)(Γ) 2k+2l+2n [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):=(12πiz) rff^{(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)sl(2)-invariant bilinear forms from old ones, so called transvectants found by Gordan,

Moscovici and Connes have constructed a sequence of Hopf algebras q\mathcal{H}_q related to geometry of foliations. Hopf algebra q\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