Quasimodular Hecke algebras and Hopf actions

Let Gamma = Gamma(N) be a principal congruence subgroup of SL2 (Z). In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra Q(Gamma) of quasimodular Hecke operators of level Gamma. Then, Q(Gamma) carries an action of ``the Hopf algebra H-1 of codimension 1 foliations'' that also acts on the modular Hecke algebra A(Gamma) of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra Q(Gamma). Further, for each sigma is an element of SL2(Z), we introduce the collection Q(sigma)(Gamma) of quasimodular Hecke operators of level Gamma twisted by sigma. Then, Q(sigma)(Gamma) is a right Q(Gamma)-module and is endowed with a pairing (_,_): Q(sigma)(Gamma) circle times Q(sigma)(Gamma) -> Q(sigma)(Gamma) We show that there is a ``Hopf action'' of a certain Hopf algebra h(1) on the pairing on Q(sigma)(Gamma). Finally, for any sigma is an element of SL2(Z), we consider operators acting between the levels of the graded module Q(sigma)(Gamma) = circle plus(m is an element of Z)Q(sigma)(m)(Gamma), where sigma(m) = GRAPHICS] . sigma for any m is an element of Z. The pairing on Q(sigma)(Gamma) can be extended to a graded pairing on Q(sigma)(Gamma) and we show that there is a Hopf action of a larger Hopf algebra H-z superset of H-1 on the pairing on Q(sigma)(Gamma).