Quasimodular Hecke algebras and Hopf actions

Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$. In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra $\mathcal Q(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$. Then, $\mathcal Q(\Gamma)$ carries an action of "the Hopf algebra $\mathcal H_1$ of codimension $1$ foliations" that also acts on the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra $\mathcal Q(\Gamma)$. Further, for each $\sigma\in SL_2(\mathbb Z)$, we introduce the collection $\mathcal Q_\sigma(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$ twisted by $\sigma$. Then, $\mathcal Q_\sigma(\Gamma)$ is a right $\mathcal Q(\Gamma)$-module and is endowed with a pairing $(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_\sigma(\Gamma)$. We show that there is a "Hopf action" of a certain Hopf algebra $\mathfrak{h}_1$ on the pairing on $\mathcal Q_\sigma(\Gamma)$. Finally, for any $\sigma\in SL_2(\mathbb Z)$, we consider operators acting between the levels of the graded module $\mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\oplus}\mathcal Q_{\sigma(m)}(\Gamma)$, where $\sigma(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma$ for any $m\in \mathbb Z$. The pairing on $\mathcal Q_\sigma(\Gamma)$ can be extended to a graded pairing on $\mathbb Q_\sigma(\Gamma)$ and we show that there is a Hopf action of a larger Hopf algebra $\mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1$ on the pairing on $\mathbb Q_\sigma(\Gamma)$.