Hopf action and Rankin-Cohen brackets on an Archimedean complex
Summary: The Hopf algebra \(\mathcal H_1\) of ``codimension 1 foliations'', generated by operators \(X, Y\) and \(\delta_n, n\geq 1\), satisfying certain conditions, was introduced by \textit{A. Connes} and \textit{H. Moscovici} [Commun. Math. Phys. 198, No. 1, 199--246 (1998; Zbl 0940.58005)]. In [Mosc. Math. J. 4, No. 1, 67--109 (2004; Zbl 1122.11023)], it was shown that, for any congruence subgroup \(\Gamma\) of \(\mathrm{SL}_2(\mathbb Z)\), the action of \(\mathcal H_1\) on the ``modular Hecke algebra'' \(\mathcal A(\Gamma)\) captures classical operators on modular forms. In this paper, we show that the action of \(\mathcal H_1\) captures the monodromy and Frobenius actions on a certain module \(\mathbb B^*(\Gamma)\) that arises from the Archimedean complex of \textit{C. Consani} [Compos. Math. 111, No. 3, 323--358 (1998; Zbl 0932.14011)]. The object \(\mathbb B^*(\Gamma)\) replaces the modular Hecke algebra \(\mathcal A(\Gamma)\) in our theory. We also introduce a ``restricted'' version \(\mathbb B^*_r(\Gamma)\) of the module \(\mathbb B^*(\Gamma)\) on which the operators \(\delta_n, n\geq 1\), of the Hopf algebra \(\mathcal H_1\) act as zero. Thereafter, we construct Rankin-Cohen brackets of all orders on \(\mathbb B^*_r(\Gamma)\).
scientific article; zbMATH DE number 5959390
scientific article