Rankin-Cohen bracket

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 $f^{(r)} := \left(\frac{1}{2\pi i}\frac{\partial}{\partial z}\right)^r f$. 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.

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Last revised on March 30, 2021 at 20:30:27. See the history of this page for a list of all contributions to it.