# Hopf action and Rankin–Cohen brackets on an Archimedean complex

### Abhishek Banerjee

The Ohio State University, Columbus, USA

## Abstract

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 Connes and Moscovici in [1]. In [2], it was shown that, for any congruence subgroup $\Gamma$ of 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 Consani [4]. 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 $\mathbb B^*_r(\Gamma)$.