The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Journal of Noncommutative Geometry


Full-Text PDF (366 KB) | Metadata | Table of Contents | JNCG summary
Volume 12, Issue 3, 2018, pp. 1041–1080
DOI: 10.4171/JNCG/297

Published online: 2018-10-30

Quasimodular Hecke algebras and Hopf actions

Abhishek Banerjee[1]

(1) Indian Institute of Science, Bangalore, India

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

Keywords: Modular Hecke algebras, Hopf actions

Banerjee Abhishek: Quasimodular Hecke algebras and Hopf actions. J. Noncommut. Geom. 12 (2018), 1041-1080. doi: 10.4171/JNCG/297