Hopf action and Rankin-Cohen brackets on an Archimedean complex
type: Item
instance of: scholarly article
title: Hopf action and Rankin-Cohen brackets on an Archimedean complex
zbMATH Open document ID: 1263.11051
DOI: 10.4171/JNCG/81
author: Abhishek Banerjee
published in: Journal of Noncommutative Geometry
publication date: 2011-10-17T00:00:00Z
review text: Summary: The Hopf algebra \(\mathcal H_1\) of ``codimension 1 foliations'', generated by operators \(X, Y\) and \(\delta_n, n\geq 1\), satisfying certain conditions, was introduced by \textit{A. Connes} and \textit{H. Moscovici} [Commun. Math. Phys. 198, No. 1, 199--246 (1998; Zbl 0940.58005)]. In [Mosc. Math. J. 4, No. 1, 67--109 (2004; Zbl 1122.11023)], it was shown that, for any congruence subgroup \(\Gamma\) of \(\mathrm{SL}_2(\mathbb Z)\), the action of \(\mathcal H_1\) on the ``modular Hecke algebra'' \(\mathcal A(\Gamma)\) captures classical operators on modular forms. In this paper, we show that the action of \(\mathcal H_1\) captures the monodromy and Frobenius actions on a certain module \(\mathbb B^*(\Gamma)\) that arises from the Archimedean complex of \textit{C. Consani} [Compos. Math. 111, No. 3, 323--358 (1998; Zbl 0932.14011)]. The object \(\mathbb B^*(\Gamma)\) replaces the modular Hecke algebra \(\mathcal A(\Gamma)\) in our theory. We also introduce a ``restricted'' version \(\mathbb B^*_r(\Gamma)\) of the module \(\mathbb B^*(\Gamma)\) on which the operators \(\delta_n, n\geq 1\), of the Hopf algebra \(\mathcal H_1\) act as zero. Thereafter, we construct Rankin-Cohen brackets of all orders on \(\mathbb B^*_r(\Gamma)\).
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 16T05
zbMATH DE Number: 5959390
zbMATH Keywords: Hopf algebras
zbMATH Keywords: modular forms
zbMATH Keywords: archimedean cohomology
MaRDI profile type: Publication
cites work: Hopf algebras, cyclic cohomology and the transverse index theorem
cites work: Q4655966
cites work: Q4655967
cites work: Q4391358
cites work: Noncommutative geometry, dynamics, and \(\infty\)-adic Arakelov geometry
cites work: Q5488005
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Modular forms and differential operators
full work available at URL: https://doi.org/10.4171/jncg/81
OpenAlex ID: W2034566453
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description: scientific article; zbMATH DE number 5959390