Quasimodular Hecke algebras and Hopf actions

Abstract

Let $\Gamma =\Gamma (N)$ be a principal congruence subgroup of $S{L}_2(\mathbb{Z})$. 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 $\sigma \in S{L}_2(\mathbb{Z})$, 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

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 $\sigma \in S{L}_2(\mathbb{Z})$, 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

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 ${\mathbb{Q}}_{\sigma }(\Gamma )$.