Quasimodular Hecke algebras and Hopf actions

  • Abhishek Banerjee

    Indian Institute of Science, Bangalore, India

Abstract

Let be a principal congruence subgroup of . In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra of quasimodular Hecke operators of level . Then, carries an action of "the Hopf algebra of codimension 1 foliations" that also acts on the modular Hecke algebra of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra . Further, for each , we introduce the collection of quasimodular Hecke operators of level twisted by . Then, is a right -module and is endowed with a pairing

We show that there is a "Hopf action" of a certain Hopf algebra on the pairing on . Finally, for any , we consider operators acting between the levels of the graded module , where

for any . The pairing on can be extended to a graded pairing on and we show that there is a Hopf action of a larger Hopf algebra on the pairing on .

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