# Quasimodular Hecke algebras and Hopf actions

### Abhishek Banerjee

Indian Institute of Science, Bangalore, India

## Abstract

Let $Γ=Γ(N)$ be a principal congruence subgroup of $SL_{2}(Z)$. In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra $Q(Γ)$ of quasimodular Hecke operators of level $Γ$. Then, $Q(Γ)$ carries an action of “the Hopf algebra $H_{1}$ of codimension 1 foliations” that also acts on the modular Hecke algebra $A(Γ)$ of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra $Q(Γ)$. Further, for each $σ∈SL_{2}(Z)$, we introduce the collection $Q_{σ}(Γ)$ of quasimodular Hecke operators of level $Γ$ twisted by $σ$. Then, $Q_{σ}(Γ)$ is a right $Q(Γ)$-module and is endowed with a pairing

We show that there is a "Hopf action" of a certain Hopf algebra $h_{1}$ on the pairing on $Q_{σ}(Γ)$. Finally, for any $σ∈SL_{2}(Z)$, we consider operators acting between the levels of the graded module $Q_{σ}(Γ)=m∈Z⊕ Q_{σ(m)}(Γ)$, where

for any $m∈Z$. The pairing on $Q_{σ}(Γ)$ can be extended to a graded pairing on $Q_{σ}(Γ)$ and we show that there is a Hopf action of a larger Hopf algebra $h_{Z}⊇h_{1}$ on the pairing on $Q_{σ}(Γ)$.

## Cite this article

Abhishek Banerjee, Quasimodular Hecke algebras and Hopf actions. J. Noncommut. Geom. 12 (2018), no. 3, pp. 1041–1080

DOI 10.4171/JNCG/297