The Hopf algebra of “codimension 1 foliations”, generated by operators , and , , satisfying certain conditions, was introduced by Connes and Moscovici in . In , it was shown that, for any congruence subgroup of SL, the action of on the “modular Hecke algebra” captures classical operators on modular forms. In this paper, we show that the action of captures the monodromy and Frobenius actions on a certain module that arises from the Archimedean complex of Consani . The object replaces the modular Hecke algebra in our theory. We also introduce a “restricted” version of the module on which the operators , , of the Hopf algebra act as zero. Thereafter, we construct Rankin–Cohen brackets of all orders on .
Cite this article
Abhishek Banerjee, Hopf action and Rankin–Cohen brackets on an Archimedean complex. J. Noncommut. Geom. 5 (2011), no. 3, pp. 401–421DOI 10.4171/JNCG/81