Quasimodular Hecke algebras and Hopf actions

Abstract

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