JournalsjncgVol. 5 , No. 3DOI 10.4171/jncg/81

Hopf action and Rankin–Cohen brackets on an Archimedean complex

  • Abhishek Banerjee

    The Ohio State University, Columbus, USA
Hopf action and Rankin–Cohen brackets on an Archimedean complex cover

Abstract

The Hopf algebra H1\mathcal H_1 of “codimension 1 foliations”, generated by operators XX, YY and δn\delta_n, n1n\geq 1, satisfying certain conditions, was introduced by Connes and Moscovici in [1]. In [2], it was shown that, for any congruence subgroup Γ\Gamma of SL2(Z)_2(\mathbb Z), the action of H1\mathcal H_1 on the “modular Hecke algebra” A(Γ)\mathcal A(\Gamma) captures classical operators on modular forms. In this paper, we show that the action of H1\mathcal H_1 captures the monodromy and Frobenius actions on a certain module B(Γ)\mathbb B^*(\Gamma) that arises from the Archimedean complex of Consani [4]. The object B(Γ)\mathbb B^*(\Gamma) replaces the modular Hecke algebra A(Γ)\mathcal A(\Gamma) in our theory. We also introduce a “restricted” version Br(Γ)\mathbb B^*_r(\Gamma) of the module B(Γ)\mathbb B^*(\Gamma) on which the operators δn\delta_n, n1n\geq 1, of the Hopf algebra H1\mathcal H_1 act as zero. Thereafter, we construct Rankin–Cohen brackets of all orders on Br(Γ)\mathbb B^*_r(\Gamma).