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Journal of Noncommutative Geometry

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Volume 9, Issue 4, 2015, pp. 1155–1173
DOI: 10.4171/JNCG/219

Published online: 2016-01-06

Action de Hopf sur les opérateurs de Hecke modulaires tordus

Abhishek Banerjee[1]

(1) Indian Institute of Science, Bangalore, India

Let $\Gamma \subseteq SL_2(\mathbb Z)$ be a principal congruence subgroup. For each $\sigma \in SL_2(\mathbb Z)$, we introduce the collection $\mathcal A_\sigma (\Gamma)$ of modular Hecke operators twisted by $\sigma$. Then, $\mathcal A_\sigma(\Gamma)$ is a right $\mathcal A (\Gamma)$-module, where $\mathcal A (\Gamma)$ is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra $\mathfrak{h}_0$ on $\mathcal A_\sigma (\Gamma)$, we define reduced Rankin–Cohen brackets on $\mathcal A_\sigma (\Gamma)$. Moreover $\mathcal A_\sigma (\Gamma)$ carries an action of $\mathcal H_1$, where $\mathcal H_1$ is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels $\mathcal A_\sigma (\Gamma)$, ${\sigma\in SL_2(\mathbb Z)}$. We show that the action of these operators can be expressed in terms of a Hopf algebra $\mathfrak{h}_{\mathbb Z}$.

Keywords: Modular Hecke algebras, Rankin–Cohen brackets, Hopf actions

Banerjee Abhishek: Action de Hopf sur les opérateurs de Hecke modulaires tordus. J. Noncommut. Geom. 9 (2015), 1155-1173. doi: 10.4171/JNCG/219