Quasimodular Hecke algebras and Hopf actions

TL;DR

This paper extends the theory of modular Hecke algebras to quasimodular forms, introduces new operators, and explores Hopf algebra actions on these structures, revealing new symmetries and module interactions in the context of level $\Gamma$ and twisted operators.

Contribution

The paper defines quasimodular Hecke algebras, introduces new operators, and establishes Hopf algebra actions on these algebras and their pairings, extending prior modular Hecke algebra theory.

Findings

Defined the algebra of quasimodular Hecke operators $\\mathcal Q(\Gamma)$.

Established Hopf algebra actions on quasimodular Hecke algebras and pairings.

Constructed graded modules and extended Hopf actions to larger algebraic structures.

Abstract

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 $\mathbb Q_\sigma(\Gamma)$.